Predict, Refine, Synthesize: Self-Guiding Diffusion Models for Probabilistic Time Series Forecasting
Authors: Marcel Kollovieh
Abdul Fatir Ansari
Michael Bohlke-Schneider
Jasper Zschiegner
Hao Wang
Bernie Wang
Venue: NeurIPS 2023 poster (2023)
Keywords: diffusion models
time series forecasting
generative modeling
deep learning
TL;DR: We propose TSDiff, an unconditional diffusion model for time series, and two inference schemes to condition it for forecasting tasks, without requiring auxiliary networks.

Abstract: Diffusion models have achieved state-of-the-art performance in generative modeling tasks across various domains. Prior works on time series diffusion models have primarily focused on developing conditional models tailored to specific forecasting or imputation tasks. In this work, we explore the potential of task-agnostic, unconditional diffusion models for several time series applications. We propose TSDiff, an unconditionally-trained diffusion model for time series. Our proposed self-guidance mechanism enables conditioning TSDiff for downstream tasks during inference, without requiring auxiliary networks or altering the training procedure. We demonstrate the effectiveness of our method on three different time series tasks: forecasting, refinement, and synthetic data generation. First, we show that TSDiff is competitive with several task-specific conditional forecasting methods (*predict*). Second, we leverage the learned implicit probability density of TSDiff to iteratively refine the predictions of base forecasters with reduced computational overhead over reverse diffusion (*refine*). Notably, the generative performance of the model remains intact — downstream forecasters trained on synthetic samples from TSDiff outperform forecasters that are trained on samples from other state-of-the-art generative time series models, occasionally even outperforming models trained on real data (*synthesize*).

Our code is available at https://github.com/amazon-science/unconditional-time-series-diffusion

Predict, Refine, Synthesize: Self-Guiding Diffusion Models for Probabilistic Time Series Forecasting
Marcel Kollovieh
Abdul Fatir Ansari
Michael Bohlke-Schneider"
Jasper Zschiegner'
Hao Wang'
Yuyang Wang'
!AWS AI Labs
Technical University of Munich
Correspondence to: ansarnd0amazon.de
Abstract
Diffusion models have achieved state-of-the-ar performance in generative mod- eling tasks across various domains. Prior works time series diffusion mod- els have primarily focused on developing conditional models tailored to specific forecasting or imputation tasks. In this work, we explore the potential of task- agnostic, unconditional diffusion models for several time series applications. We propose TSDiff, an unconditionally trainned diffusion model for time series. Our proposed self-guidance mechanism enables conditioning TSDiff for downstream tasks during inference, without requiring auxiliary networks or altering the training procedure. We demonstrate the effectiveness of our method on three different time series tasks: forecasting, refinement, and synthetic data generation. First, we show that TSDiff is competitive with several task-specific conditional forecasting methods (predict). Second, we leverage the learned implicit probability density of TSDiff iteratively refine the predictions of base forecasters with reduced computational overhead over reverse diffusion (refine.. Notably, the generative performance of the model remains intact downstream forecasters trained on synthetic samples from TSDiff outperform forecasters that are trained on sam- ples from other tate-offthee-rr generative time series models, occasionally even outperforming models trained on real data (synthesize).
Introduction
Time series forecasting informs key business decisions [27], for example in finance [47], renewable energy [56], and healthcare [7]. Recently, deep learning- based models have been successfully applied to the problem of time series forecasting [46, 5]. Instantiations of these methods use, among other techniques, autoregressive modeling [46, 42], to-suuuee----eeeeeee modeling [54], and normalizing
flows [43, 8]. These techniques view forecasting the problem of conditional generative modeling: generate the future, conditioned on the past.
Diffusion models [48, 20] have shown outstanding performance on generative tasks across various domains [10, 28, 30, 36, 44] and have quickly become the framework of choice for generative modeling. Recent studies used conditional diffusion models for time series forecasting and imputation tasks [44, 52, 1, 6]. However, these models are task specific, i.c., their applicability is limited to the specifie imputation or forecasting task they have been trained Consequently, they also forego the desirable unconditional generative capabilities diffusion models.
Equual contribution.
Work conducted during an internship Amazon.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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(o(
Po(F))
F)(y:))
Synthiic Samphia
Vile
Via
Train
Nix (p
30
Y0:
vi6)
Downetheam
Predict
pPa MM
Predict
Refine
Synthesize
Figure I: An overview of TSDiff's use cases. Predict: By utilizing observation self-guidance, TSDiff can be conditioned during inference to perform predictive tasks such as forecasting (see Sec. 3.1). Refine: Predictions of base forecasters can be improved by leveraging the implicit probability density of TSDiff (see Sec. 3.2). Synthesize: Realistic samples generated by TSDiff can be used to train downstream forecasters achieving good performance on test data (see Sec. 4.3).
This raises natural research question: Can we address multiple (even conditional) downstream tasks with unconditional diffusion model? Specifically, investigate the usability of task-agnostic unconditional diffuuion models for forecasting tasks. We introduce TSDiff, an unconditional diffusion model for time series, and propose two inference schemes to utilize the model for forecasting, Build- ing upon recent work on guided diffusion models [10, 19], we propose self-guidance mechanism that enables conditioning the model during inference, without requiring auxiliary networks. This makes the unconditional model amenable to arbitrary forecasting (and imputation) tasks that are conditional in nature". We conducted comprehensive experiments demonstrating that our self-guidance approach competitive against task-specific models on several datasets and across multiple forecasting scenar- ios, without requiring conditional training. Additionally, we propose method to iteratively refine predictions of base forecasters with reduced computational overhead compared to reverse diffusion by interpreting the implicit probability density learned by TSDiff as an energy-based prior. Finally, we show that the generative capabilities of TSDiff remain intact. We train multiple downstream forecasters on synthetic samples from TSDiff and show that forecasters trained on samples from TSDiff outperform those trained on samples from variational autoencoders [9] and generative adver- sarial networks [57], sometimes even outperforming models trained on real samples. To quantify the generative performance, we introduce the Linear Predictive Score (LPS) which we define as the test forecast performance of a linear ridge regression model trained on synthetic samples. TSDiff significantly outperforms competing generative models in terms of the LPS several benchmark datasets. Fig. highlights the three use cases of TSDiff: predict, refine, and synthesize.
In summary, our key contributions are:
TSDiff, unconditionally trained diffusion model for time series and mechanism condition TSDiff during inference for arbitrary forecasting tasks (observation self ceffguuidnce);;
An iterative scheme to refine predictions from base forecasters by leveraging the implicit probability density Icarned by TSDiff:
Experiments on multiple benchmark datasets and forecasting scenarios demonstrating that observation self-guid -efddnuidence is competitive against task-specific conditional baselines; Linear Predictive Score, metric to evaluate the predictive quality of synthetic samples, and experiments demonstrating that TSDiff generates realistic samples that outperform competing generative models in terms of their predictive quality.
The rest of the paper is organized as follows. Sec. 2 introduces the relevant background on denoising diffusion probabilistic models (DDPMs) and diffusion guidance. In Sec. 3, we present TSDiff, an unconditional diffusion model for time series, and propose two inference schemes to utilize the model for forecasting tasks. Sec. discusses the related work on diffusion models for time series and diffusion guidance. Our empirical results are presented in Sec. 4. We conclude with summary of our findings, the limitations of our proposals, and their potential resolutions in Sec. 6.
'Note that in contrast to meta-learning and foundation models literature, we train models dataset and only vary the inference task, e.g., forecasting with different missing value scenarios.

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2 Background
2.1 Denoising Diffusion Probabilistic Models
Diffusion models [48, 20] provide a framework for modeling the data generative process as discrete- time diffusion process. They are latent variable models of the form pe (y) pe (y, x xI:T ) 1:7 where y ~ q(y) is the true underlying distribution. The latent variables {x .....''' are generated by a fixed Markov process with Gaussian transitions, often referred to as the forward process,
T x- x = y) II x(x")x/=1) and p(x²)x(-1)===00
q(x
1, (1)
where Bg is the variance of the additive noise, is the observed datapoint, and g(x²) (O) The fixed Gaussian forward process allows direct sampling from g(x"ly),
Vary + v(1 (t,),,
where a₁ 1- - 3r, a. = II as and (.. On the other hand, the reverse diffusion (generative) process is formulated
T
Po (x = y, y...., x²) = p(x' 7)
II [=1
pa px
[x*) and >xxx"=(xx') == NNua (x),xx),),)))) ozl),
(3)
where p(x") ~ (O,1) and or = I-
T- 3. The model is trained to approximate the true reverse
process - x²) by maximizing approximation of the evidence lower bound (ELBO) of the log-likelihood. Specifically, μl parameterized using a denoising network, c0.
Se
( t)
x
Ex(x',
(4)
Voa
V1 āt
which is trained to predict the sampled noise (c in Eq. 2) using the simplified objective function [20],
Ey || () el|?]
By suitably adjusting the denoising neural network incorporate the conditioning input, this objective can be employed to train both unconditional and conditional models.
2.2 Diffusion Guidance
Classifier guidance repurposes unconditionalll-triin trained image diffusion models for class-conditional image generation [10]. The key idea constitutes decomposing the class-conditional score function using the Bayes rule,
Vx 10o'((''le (c) = vx logp(x') + Vxi logp(c|x*),
and employing an auxiliary classifier estimate Vx log p(c|x'). Specifically, the following modified reverse diffusion process (Eq. 3) allows sampling from the class-conditioms distribution,
Po (x' |x c) <Xxx'=;; "ox'"";"ox'" () +
V Vx log pog'(('x'),
where is a scale parameter controlling the strength of the guidance.
3 TSDiff: an Unconditional Diffusion Model for Time Series
In this section, we present main contributions: TSDiff, unconditional diffusion model designed for time series, along with two inference schemes that leverage the model for downstream forecasting tasks. We begin by outlining the problem setup and providing concise overview of our network architecture. Subsequently, we delve into our first scheme observation self-guidance - which enables conditioning reverse diffusion arbitrary observed timesteps during inference. Secondly, we present a technique to iteratively refine predictions of arbitrary base forecasters by utilizing the implicit probability density Iearned by TSDiff as prior.

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Problem Statement. y E R' be time series of length L. Denote obs C fl..... L} as the of observed timesteps and ta as its complement of target timesteps. Our goal to recover the complete time series y, given the observed subsequence Yohs which may or may not be contiguous. Formally, this involves modeling the conditional distribution po (y Yobs). This general setup subsumes forecasting tasks, with or without missing values, as special cases. We seek to train a single unconditional generative model, pe (y), and condition it during inference to draw samples from arbitrary distributions of interest, po (Yta Nobs).
Generative Model Architecture. We begin with modeling the marginal probability, po(y), via diffusion model, referred TS- Diff, parameterized by 9. The ar- chitecture of TSDiff is depicted in Fig. 2 and is based on SSSD [1] which is a modification of Dif- fWave [28] employing S4 lay- ers [18]. TSDiff is designed to han- dle univariate sequences of length L. To incorporate historical infor- mation beyond L timesteps with- out increasing L, we append lagged time series along the channel di- mension. This results in a noisy input x' R Lx (see Eq. 2) to the diffusion model, where C - the number of lags. The $4 layers oper- ate on the time dimension whereas the Convix1 layers operate on the channel dimension, facilitating information flow along both dimensions. As typical for unconditional diffusion models, the output dimensions of TSDiff match the input dimensions. Note that while we focus on univariate time series in this work, TSDiff can be modified handle multivariate time series by incorporating additional layers, e.g., a transformer layer, operating across the feature dimensions after the S4 layer.
wm
Observation Sel Oublance
T5c0e Archilactrre
whm
Swin X
Akywer
Figure 2: An overview of observation self-guidance, The predicted noise, es (x', t). first denoises x" unconditionally as x'-1 and approx- imates y as ÿ. The reverse diffusion step then guides x'-1 via the log-likeldhood of the observation Yohs under a distribution parameter- ized by y.
In the following, discuss two approaches to condition the generative model, pe (y). during inference, enabling us to draw samples from pe (Ytalyobs) Yobs).
3.1 Observation Self-Guidance
Let f 2 0 be arbitrary diffusion step. Applying Bayes' rule, have,
(x yybb)) x (x
(8)
which yields the following relation between the conditional and marginal score functions,
Vx log Po (xm))) = Vx log po (Yobs x²) + Vx log')(x')..
(9)
Given access to the guidance distribution, pa Pe(yobalx') (x',, we can draw samples from (y (obs) using guided reverse diffusion, akin Eq. (7),
po (x |x' yobs) Nxx :1 t) V log log (xr_oo/oboxx))))))))))) xx),,,1)) I.)..
(10)
The scale parameter 8 controls how strongly the observations, Yobs and the corresponding timesteps in the diffused time series, y. align. Unlike Dhariwal and Nichol [10], we do not have access to auxiliary guidance networks. In the following, we propose two variants of a self-guidance mechanism that utilizes the same diffusion model to parameterize the guidance distribution. The main intuition behind self-guidance that a model designed for complete sequences should reasonably approximate partial sequences. A pseudo-code of the observation self-guidance is given in App. A.2.
Mean Square Self-Guidance.
We model po (ybbe//'' as multivariate Gaussian distribution,
Po (x') Ny fo(x' tt, I),
(11)

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where fo is function approximating y, given the noisy time series x'. We can reuse the denoising network Eg to estimate y as
ÿ ( x' v (1-az (oo
(12)
which follows by rearranging Eq. (2) with e Eo(x', f), as shown by Song et al. [49]. This one-step denoising serves cost-effective approximation of the model for the observed time series and provides the requisite guidance term in the form of the score function, Vx: log pe (Yobs) (ooxxx)),. which can be computed by automatic differentiation. Consequently, our self-guidance approach requires no auxiliary networks or changes to the training procedure. Applying the logarithm to Eq. (11) and dropping constant terms yields the mean squared error (MSE) loss on the observed part of the time series, hence we named this technique mean square self seff-guidncee.
Quantile Self-Guidance. Probabilistic forecasts are often evaluated using quantile-based metrics such as the continuous ranked probability score (CRPS) [15]. While the MSE only quantifies the average quadratic deviation from the mean, the CRPS takes all quantiles of the distribution into account by integrating the quantile loss (also known as the pinball loss) from 0 to 1. This motivated us substitute the Gaussian distribution with the asymmetric Laplace distribution that has been studied in the context of Bayesian quantile regression [58]. The probability density function of the asymmetric Laplace distribution is given by,
Po (Yoba xx) Z exp (-3 max {K (Yohe fa(xx,)), 1)- (Yobs fo(x( (13)
where Z is normalization constant, 6 > 0 a scale parameter, and (0, 1) an asymmetry parameter. Setting b = 1, the log density yields the quantile loss with the score function,
xx log peppee (Yobs Vx max(x (Yobe - fo('', t)), (N 1) -yobs fo(x', (e(x((t))3 f))}
(14)
with x specifying the quantile level. By plugging Eq. (14) into Eq. (9), the reverse diffusion can be guided towards specific quantile level s. practice, we use multiple evenly spaced quantile levels in (0, 1), based on the number of samples in the forecast. Intuitively, we expect quantile self- self-guidance
to generate more diverse predictions by better representing the cumulative distribution function.
3.2 Prediction Refinement
In the previous section, we discussed a technique enabling the unconditional model to generate predictions by employing diffusion guidance. In this section, we will discuss another approach repurposing the model to refine predictions of base forecasters. Our approach is completely agnostic to the type of base forecaster and only assumes access to forecasts generated by them. The initial forecasts are iteratively refined using the implicit density learned by the diffusion model which serves prior. Unlike reverse diffusion which requires sequential sampling of all latent variables, refinement is performed directly in the data space. This provides a trade-off between quality and computational overhead, making it an economical alternative when the number of refinement iterations is less than the number of diffusion steps. Furthermore, in certain industrial forecasting scenarios, one has access a complex production forecasting system of black-box nature. In these cases, refinement presents a cost-effective solution that enhances forecast accuracy post hoc, without modifying the core forecasting process- a change that could potentially be lengthy procedure.
In the following, we present two interpretations of refinement as (a) sampling from an energy function, and (b) maximizing the likelihood to find the most likely sequence.
Energy-Bass Sampling. Recall that our goal is to draw samples from the distribution p(yialyan Yote). Let be an arbitrary base forecaster and g(yobs) be sample forecast from g which serves as initial guess of a sample from Yobse. To improve this initial guess, we formulate refinement as the problem of sampling from the regularized energy- based model (EBM).
(15) where y is the time series obtained upon combining y Yobs and g(yons), and R is a regularizer such as the MSE loss or the quantile loss. We designed the energy function such that low energy is assigned to samples that are likely under the diffusion model, po(y), and also close to ý, ensured by the diffusion log- le-eliikiioodd and the regularizer in the energy function, respectively. The Lagrange
y) log XR(y,
5

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multiplier, X, may be tuned to control the strength of regularization; however, we set it to 1 in our experiments for simplicity.
We use overlamped Langevin Monte Carlo (LMC) [50] to sample from this EBM. ((o) is initialized to y and iteratively refined as,
((++)) - yo V/ yoo ()()) + 22m and & & &,O,),
(16)
where 7) and y the step size and noise scale, respectively.
Note that in contrast observation self- guidance, we directly refine the time series in the data space and require an initial forecast from base forecaster. However, similar to observation self- -if-dguiance,, this approach does not require any modifications to the training procedure and can be applied to any trained diffusion model. A pseudo-code of the energy-based refinement is provided in App. A.2.
Maximizing the I Likelihood. The decomposition in Eq. (15) can also be interpreted as a regularized optimization problem with the goal of finding the most likely time series that satisfies certain constraints on the observed timesteps. Concretely, it translates into,
arg min [- lolog po + XR(y.5).
(17)
which can be optimized using gradient descent and is special case of Eq. (16) with y 0. Given the non-convex nature of this objective, convergence to the global optimum is not guaranteed. Therefore, expect the initial value, Yro). influence the resulting time series, which can also be observed in the experiment results (see Table 3).
Approximation of log po()) To approximate the log-likelihood log po (y) we utilize the objective used to train diffusion models,
logpe(y) 2 EEe, [xle((x', -
(18)
which is simplification of the ELBO [20]. However, a good approximation requires sampling several diffusion steps (f) incurring computational overhead and slowing down inference. To speed up inference, we propose to approximate Eq. (18) using only single diffusion step. Instead of randomly sampling/ 1, we use the representative step, т, which improved refinement stability in our experiments. The representative step corresponds to the diffusion step that best approximates Eq, (18), i.e..
arg min
[Ee,,y [xl=o(x",t)
=l - Be,y [|
-=l3])
(19)
The representative step is computed only once per dataset. can be efficiently computed post training by computing the loss at every diffusion step randomly-sampled batch of training datapoints and then finding the diffusion step closest to the average loss. Alternatively, we can keep a running average loss for each t and compute the representative step using these rumning averages. This can be done efficiently because the loss used to obtain T is the same as the training loss for the diffusion model.
Experiments
In this section, we present empirical results on several real-world datasets. Our goal is to investigate whether unconditional time series diffusion models can be employed for downstream tasks that typically require conditional models. Concretely, we tested if (a) the self-guidance mechanism in TSDiff can generate probabilistic forecasts (also in the presence of missing values), (b) the implicit probability density learned by TSDiff can be leveraged to refine the predictions of base forecasters, and (c) the synthetic samples generated by TSDiff are adequate for training downstream forecasters."
Datasets and Evaluation. We conducted experiments eight univariate time series datasets from different domains, available in Gluon'TS [2] - Solar [29], Electricity [11], Traffic [1 Exchange [29],
M4-Hourly [37]. UberTI.C-Hourly [13], KDDCup [16], and Wikipedia [14]. We evaluated the quality of probabilistic forecasts using the continuous ranked probability score (CRPS) [15]. We approximated the CRPS by the normalized average quantile loss using 100 sample paths, and report
means and standard deviations over three independent runs (see App. B for details on the datasets and evaluation metrie).
'Our code is available github. http://mmmzoosoccccoccuuunoooono www.aal--ee--ssee---iieee---

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Table 1: Forecasting results eight benchmark datasets. The best and second best models have been shown as bold and underlined, respectively.
Mathodd
Solar
Electricity
Teuffic
Eechange
K4-Hourly
Ubee:UC-Bouurly
KDDCup
Wikiipedia
Seanonal Madre
0.512100.00
0:0:9100000
0,201:100000
0.001:00.000
0048110.000
0,299:10.000
0:000:10000
0,410:10.000
ARIMA
0.545=00.005
0.0080000000
00000000011
0.2644-00001
0.347:000003
RTS
0:61==0000
0.77:000044
0433=0000
0002000001
6422+00001
0.751:200088
0.71++0007
Lineur
0.509=00.021
0.080+000008
0.179
0.01 00.001
00000000000
0. 0.500
0.513=00.011
1.624=1.144
DeapAR
0.389=0.001
0154-000000
00095000001
0
01052000006
0.161=000002
0.1442000077
0.231 0000K
XQ-OHH
0.790=00.03
0000
0.01900.000
0.06:00000
0.516:00012
0:200=30001
DeepSistto
0.379=00001
0:075200004
0:14::0011
0.011+0.001
00,022
0.258=00.087
0:318=00011
Transformer
0,419 0.009
0.00610.018
0.102.00.022
0.01010.000
004010,0014
0.192.1.0004
0.00110:021
0.21411.001
TPT
0.417=00.025
0.066+000008
0.1342:0.077
0.007=001000
01009:0000011
0. 0.195:000006
000055
0.229=00006
0.352.20.005 005
0054:100000
0:150:000022
0,00320.014
0040:00.003
0:206:200002
0.315:20:022
0:289:00017
TEDLEI Coad
0.014:00.014
0.050:00002
0.0942-0003
0,01320.002
0.29900:000
0.1722-00005
0.754:10.007
0218:100010
THD: F1-M8
0.301=00.003
0.062:00001
0.116=00001
0.018+00.00
0.45++00000
0.153=000077
0. 0.325=00008
0.257=00001
TED1E1
0,358.10.020
0,08910,000
0.08810.002
1.0001
0..6610.001
0.172100005
0.311.0.026
0.201
4.1 Forecasting using Observation Self-Guidance
We tested the forecasting performance of the two proposed variants of observation self-guidance. mean square guidance (TSDifT-MS) and quantile guidance (TSDiff-Q). against several forecast- ing baselines. We included Seasonal Naive, ARIMA, ETS, and Linear (ridge) regression model from the statistical literature [21]. Additionally, we compared against deep learning mod- els that represent various architectural paradigms such as the RNN-based DeepAR [46], the CNN-based MQ-CNN [54], the state space model-based DeepState [42], and the self-attention- based Transformer [53] and TFT [32] models. We also compared against two conditional dif- conditional version of TSDiff closely related to SSSD [1] (see App. B.4 for a more in-depth discussion on the baselines). Note that we did not seek to obtain sfte-of-theear forecasting results the datasets studied but to demon- strate the efficacy of unconditional diffusion models against task-specific conditional models.
fusion models, CSDI [52] and TSDiff-Cond,
Table 1 shows that TSDiff-Q is competitive with of-teoof-the-an conditional models, but does not require task task-specific training, achieving the low-
Grund Trith Median Forecust
Femeaatt liaroon
92% Intenval
200
Eletniyy
est or second-lowest CRPS on 5/8 datasets. We also observe that the choice of guidance dis- tribution is critical. While Gaussian (TSDiff- MS) yields reasonable results, using the asym- metric Laplace distribution (TSDiff-Q) lowers the CRPS further. We hypothesize that taking the different quantile levels into account during guidance improves the results on the quantile- based evaluation metric (CRPS). Fig. 3 shows example forecasts from TSDiff-Q. Q. Note that as self-guidance only imposes a soft-constraint the observed timesteps, the resultant diffused values are not guaranteed to match the obser- vations for these timesteps. The alignment be- tween predictions and observations can be con- trolled via the scale parameter, (see Eq. 9). In practice we observed that the diffused values were close to the observations for our selected values of s. as shown in Fig. 5 (Appendix).
159
27
28
29
30
31
200
100
25
Mar
Exchenge
1.50
2013
Figure 3: Example forecasts generated by TSDiff-Q for time series in Electricity. KDDCup. and Exchange three datasets with different frequencies and/or predic- tion lengths.
Forecasting with Missing Values. We evaluated the forecasting performance with missing values in the historical context during inference to demonstrate the flexibility of self guiddncce Specifically, we tested three scenarios: (a) random missing (RM), (b) blackout missing (i.e., a window with consecutive missing values) at the beginning of the context window (BM-B), and (c) blackout missing at the end of the context window (BM-E). We removed segment equal to the context window from the end of the training time series to ensure that the model is not trained on this section. During inference, masked 50% of the timesteps from this held-out context window for each scenario
7

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while the forecast horizon remained unchanged. We trained the unconditional model on complete sequences only once per dataset, without making any modifications to the objective function. As a result, the three scenarios described above are inference only for TSDiff. For comparison, we trained conditional models (TSDiff- Conid specifically on the target missingness scenarios. These models were trained to predict the missing values in the input time series, where some timesteps were masked according to the target missingness scenario (see App. B.3 for detailed discussion on the missing values experiment setup).
Table 2: Forecasting with missing values results on six benchmark datasets.
Method
Solar
Electricity
Traffic
Exchange
UberTICCHourrl
KDDCup
BM-E BM-B RM
TSDiff-Con
0.357:1:.023
0.002
000003
0.012==0.004
0.180:10.015
0.757-10.002
TSDiff-Q
0.387.10.015
0.052110.001
0.0000000004
0.013-100000
0.183-10.002
0.397.10.042
1SDiff-Cond
0.377:1:0017
0.049:10.001
0.005
0.009:0.000
0.181:10.009
0.699-10.009
T3DIFI-Q
0.387=00.019
0.001=00000
0.100
0.011=0.001
0.182:4:0004
0.441
TSD1rf
0.376::0.036
0.065
0.123::.0033
0.035=0.021
0.179:1:0003
0.819::0:03
TSDiFI-Q
0.435=001113
0.008
0.139=00.013
0.020=0.001
0.183=+0.005
0.344#0.012
The results in Table show that TSDif--Q performs competitively against task-specific conditional models, demonstrating its robustmess with respect to missing values during inference. This makes TSDiff-Q Q particularly useful in real-world applications where missing values common but the missingness scenario is not known in advance. It is noteworthy that we did not tune the guidance scale for each missingness scenario but used the same value as in the previous standard forecasting setup. We expect the results to further improve if the scale is carefully updated based on the missingness ratio and scenario.
4.2 Refining Predictions of Base Forecasters
We evaluated the quality of the implicit probability density learned by TSDiff by refining predictions from base forecasters, described in Sec. 3.2. We considered two point forecasters (Seasonal Naive and Linear) and two probabilistic forecasters (DeepAR and Transformer) as base models. We refine the forecasts generated from these base models for 20 steps, as described in Algorithm 2 (see App. B.3 for a discussion on the number of refinement steps). Similar to seff-guidance, we experimented with the Gaussian and asymmetric Laplace negative liglliics as regularizers, R, for both energy (denoted by LMC) and maximum likelihood (denoted by ML)-based refinement.
The CRPS scores before (denoted as "Base") and after refinement reported in Table 3. At least one refinement method reduces the CRPS of the base forecast on every dataset. Energy- based refinement generally performs better than ML-based refinement for point forecasters. We hypothesize that this is
Table 3: Retinement results on eight benchmark datasets. The best and second best settings have been shown as bold and underlined, respectively.
Sotting
Bolor
Electricity
Traffic
Exchange
Mc-Hourly
KDDCup
Hikipodia
Base
0,512:00:00
0.009::00000
0:221:00000
0011=-00000
0.048-00.000
01299:000000
0.410:20.100
LMC MS
0.880:10.009
0,009100004
6.126-10.011
0.01310.001
0.00011..002
0.186:00.00
0.005
0.33910.001
LMC-Q
0,051110001
0.104
0.00910.000
0.35710.001
ML-MS
0.489110.007
0.064100006
01301:00002
001510.002
0.06610.000
0:00010000
0.510:0.002
0.309:0.001
ML-Q
0.400:10.007
0.050:10.001
01355100000
0.00910.000
0.036-1-0.01
0.215 00000
0.403-00005
0.365-0.801
Rase
0:569:60.021
0.068 008
0.179:00003
001100.001
0030+00001
0:560::0000
0 0:313:00111
1624
LMC-MS
0,495-90.019
0,059:20004
0.113+00.011
0015+00001
01840+000022
0.187
0.55=000015
1.315-0.992
LMC-Q
0516100020
0.005
0.119100000
0.00910.000
0.34.10.001
0.228 010
0.34610.010
1 0002
ML.MS
0.901-000016
0.061:200005
0117+00002
0015#0.002
0.45-20.003
0:201-00.00 002
0.722200155
127.0.993
ML-Q
0.525+6.021
0.056=-0000
0.221
0010+0000
0.132--0.001
0.240-200010
0 05300-00011
1.35
Base
0.38910.001
0.004.10.000
1.00010.001
0011100001
0.02210.006
0.161110.000
0.1410.027
00.00
DeqpAR
LMC-MS
0.398:10.004
0.059 004
01111:0001
001210.001
0.40:00.000
0. 11184120000
0.227100002
LMC-Q
0.388-10.002
0.003-20:001
0.1011000001
0.010-1..011
0.035
0.00122
1.0021
0.220 0.005
ML-MS
0.00200.000
0001110006
0.115
0.01410.001
10.000
1.19810000
0.77100.034
0.2000
ML-Q
0.366.10.022
0.002:0.0011
0.09100.001
0.01010.002
0.03510.001
0.160
0.012200021
0221100.00 000
Base
0.419:50:008
0.006:50.018
0.102200.M12
0.01010.000
0.080:1000144
0.192=20.004
0.411:00621
0.214-00001
1MC-MS
0:415+0:000
0,059-00004
0.1100000
0.11=000001
0.847+00000
D. 1.185=00000 00S
462-00014
0.220=00001
0033
LMC-QQ
0,415+00000
0.101+000001
0.010-20.000
0.038+000006
0.177=41.005
0.384=0.1015
0211
0.415-00.010
0.003-5000055
0014-H0UMM
0.198-20.00
0.470=000114
ML- Q
0.413-10.008
0.009:100005
0.099100.001
0.010.10.000
0.037 000000
0.177
0.384
0.000
8

---

Table 4: Results of forecasters trained on synthetic samples from different generative models eight benchmark datasets. Best scores are shown in bold.
Generatoo
Solar
Electricity
icaffic
Exchange
N6-Sourly
UbarTLC-Bourky
320Oup
Nikippdli
Real
0.509=0.021
0,083=00000 00S
0 003
00010000000
0.139=00000 601
0.360=0.023
0,513#00011
1.624
Libel
TineVAE
0.931:0 167
0.128_00005
0,256100000
0.006:0.000
0007
0.351 0.00
L02010.00
0643-100066
TiswGAK
1,140=01,333
0.25400064
0.395:20:922
0.01 100000
0,140=00053
0.665-00 104
0
0.21 +00025
00582
0.065=0.002
0.164:00002
0.0220.000
0.045=
0.291 0.004
0.401
0.392:20013
Real
0.300=0001
0,054=00000
0.999:0000 001
00.000
0,052:0000
20.002
0.14
0230
DeePAR
TimeVAE
0.882.00012
0.000-000001
0.000110006
0,00910,000
0.01910.0000
0.278
0.621
0.40010.0112
TimeGAM
0.976.10.79
0.183_00006
0.419110:22
0,001.0.00
0.22110035
0.59110.:125
0.322 0.600100011
TEDLEX
0.478 =0.00
0.158=00001
0.129:00003
0017+00000
0,042500224
0.191=0.015
0.378
0.223
Real
0:419:00008
0,07620018
0:100:10000
00101000000
0,00220044
0,192:0.004
04111000021
0214::0001
Homst
TimeVAR:
0:5
0007
0.161:00188
0.01 1000000000
000520011
0.291 200000
1111
0481:20017
TimeGAN
0.972200667
0.182=00008
0.413:10:044
0,009 10,00100.01
0.114
0.685 1.0.41
0.622.00016
0.314:00045
PEDLEX
0.457=00008
0.156=00001
0.143=00020
0.000+000011
0J130=BURS
0,225=8.055
0,356
0.23P=0.010
because the additive noise in LMC incentivizes exploration of the energy landscape which improves the "spread" of the initial point forecast. In contrast, ML-Q refinement outperforms energy-based refinement on probabilistic base models suggesting that sampling might not be necessary for these models due to the probabilistic nature of the initial forecasts.
Refinement provides trade-off; while has a lower computational overhead than self self-guidance, its performance strongly depends on the chosen base forecaster. It substantially improves the performance of simple point forecasters (e.g.. Seasonal Naive) and yields improvements on stronger probabilistic forecasters (e.g., Transformer) as well.
4.3 Training Downstream Models using Synthetic Data
Finally, evaluated the quality of generated samples through the forecasting performance of downstream models trained on these synthetic samples. Several metrics have previously been proposed to evaluate the quality of synthetic samples [12, 57, 26, 23]. Of these, we primarily focused on predictive metrics that involve training a downstream model on synthetic samples and evaluating its performance on real data (i.e., the train synthetic-test on real setup). In the absence of standard downstream model (such as the Inception network [51] in the case of images), prior works have proposed different network architectures for the downstream model. This makes the results sensitive to architecture choice, random initialization, and even the choice of deep learning library. Furthermore, training a downstream model introduces additional overhead per metric computation. We propose the Linear Predictive Score (LPS) to address these issues. We define the LPS as the test CRPS of a linear (ridge) regression model trained on the synthetic samples. The ridge regression model is a simple, standard model available in standard machine learning libraries (c.g., scikit-learn) that can effectively gauge the predictive quality of synthetic samples. Moreover, ridge regression model is cheap to fit it can be solved in closed-form eliminating variance introduced by initialization and training.
We compared samples generated by TSDiff against those from TimeGAN [57] and ViE [9], two time series generative models from alternative frameworks. In addition to the linear model, we trained two strong downstream forecasters (DeepAR and Transformer) on synthetic samples from each generative model. The test CRPS of these forecasters presented in Table TSDiff's samples significantly outperform Time and TimeGAN in terms of the LPS showcasing their excellent predictive quality with respect to a simple (linear) forecaster. On the stronger DeepAR and Transformer forecasters. TSDiff outperforms the baselines on most of the datasets. Moreover, the scores obtained by TSDiff are reasonable when compared to those attained by downstream forecasters trained on real samples. Notably, these forecasters trained on real data had the advantage of accessing time features and lags that extend far beyond the sequence length modeled by TSDiff. These results serve evidence that TSDiff effectively captures crucial patterns within time series datasets and generates realistic samples (see App. C for qualitative comparison between real time series and those generated by TSDiff).

---

5 Related Work
Diffusion Models. Diffusson models were originally proposed for image synthesis [48, 20, 10], but have since been applied to other domains such as audio synthesis [28], protein modeling [3], and graph modeling [24]. Prior works have also studied image inpainting using diffusion models [35, 45, 25] which is problem related to time series imputation. Similar to the maximum likelihood variant of our refinement approach, Graikos al. [17] showed the utility of diffusion models as plug-am wuug--nd-plyy priors. The architecture of our proposed model, TSDiff, is based on a modification [1] of DiffWave [28] which was originally developed for audio synthesis.
Diffusion Models for Time Series. Conditional diffusion models have been applied to time series tasks such as imputation and forecasting. The first work is due to Rasul et al. [44], who proposed TimeGrad for multivariate time series forecasting featuring conditional diffusion head. Biloš et al. [6] extended TimeGrad to continuous functions and made modifications to the architecture enabling simultaneous multi-horizon forecasts. CDSI [52] is conditional diffusion model for imputation and forecasting which is trained by masking the observed time series with different strategies. SSSD [1] is a modification of CDSI that uses S4 layers [18] as the fundamental building block instead of transformers. The models discussed above [44, 52, 1] are all conditional models, i... they are trained on specific imputation or forecasting tasks. In contrast, TSDiff is trained unconditionally and conditioned during inference using diffusion guidance, making it applicable to various downstream tasks.
Diffusion Guidance. Dhariwal and Nichol [10] proposed classifier guidance to repurpose pretrained unconditional diffusion models for conditional image generation using auxiliary image classifiers. Ho and Salimans [19] eliminated the need for additional classifier by jointly training conditional and an unconditional diffusion model and combining their score functions for conditional generation. Diffusion guidance has also been employed for text-guided image generation and editing [39, 4] using CLIP embeddings [41]. In contrast to prior work, our proposed observation self-guidance does not require auxiliary networks modifications to the training procedure. Furthermore, we apply diffusion guidance to the time series domain while previous works primarily focus on images.
6 Conclusion
In this work, we proposed TSDiff, an unconditional diffusion model for time series, and self- guidance mechanism that enables conditioning TSDiff for probabilistic forecasting tasks during inference, without requiring auxiliary guidance networks or modifications to the training procedure. We demonstrated that our task-ttnostic TSDiff, in combination with self selfdguid,nee, is competitive with strong task specific baselines on multiple forecasting tasks. Additionally, we presented a refinement scheme to improve predictions of base forecasters by leveraging the implicit probability density learned by TSDiff. Finally, we validated its generative modeling capabilities by training multiple downstream forecasters on synthetic samples generated by TSDiff. Samples from TSDiff outperform alternative generative models in terms of their predictive quality. Our results indicate that TSDiff' learns important characteristics of time series datasets, enabling conditioning during inference and high-quality sample generation, offering an attractive alternative to task-specific conditional models.
Limitations and Future Work. While observation self-guidance provides an alternative approach for state-of-the-art probabilistic time series forecasting, its key limitation is the high computational cost of the iterative denoising process. Faster diffusion solvers [49, 33) would provide trade-off between predictive performance and computational overhead without altering the training procedure. Furthermore, solvers developed specifically for accelerated guided diffusion [34, 55] could be deployed to speed up sampling without sacrificing predictive performance. Forecast refinement could be further improved by better approximations of the probability density and the utilization of momentum based MCMC techniques such as Hamiltonian Monte Carlo [38] and underdamped
Langevin Monte Carlo [50]. In this work, we evaluated TSDiff in the context of probabilistic forecasting, however, it is neither limited nor tailored to it. It can be easily applied to any imputation task similar to our forecasting with missing values experiment where we only evaluated the forecasting performance. Moreover, the core idea behind observation self- gel--guidance is not limited to time series and may be applicable to other domains. Tailoring the guidance distribution appropriately, enables the use of self-guidand to a variety of inverse problems.
10

---

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14

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A Technical Details
A.1 Asymmetric Laplace Distribution
The asymmetric Laplace distribution generalization of the (symmetrie) Laplace distribution and can be viewed two exponential distributions with unequal rates glued about location parameter. An alternative parameterization of this distribution has been employed for Bayesian quantile regression [58].
po(z) x exp (-7 max {N (z - m), 1) (2)))))))
where x is the quantile level, m the location parameter, and is the scale parameter. Setting к. = 0.5 yields the symmetric Laplace distribution. The negative log- likelihood of the asymmetric Laplace distribution corresponds to the pinball loss. An illustration of the probability density function (PDF) and the unnormalized negative bog-likelihood (NLL) for different quantile levels is shown in Fig. 4.
2-
4
6-
0.2
2
10
0.2
0.1
0.6
0.8
1.0
0.0
0.8
1.0
8
(a)
PDF
(b) NLL
Figure Probability density function (PDF) and negative log-likelihooe (NLL) of the unnormalized Asymmetric Laplace Distribution for different quantile levels (x) with m = 0 and
A.2 Algorithms
Observation Self-Guidance. The observation self-guidance can be easily implemented using guided reverse diffusion [10]. The pseudo-code of observation self-guidance is provided in Alg. Given an observation, Yobs, and guidance scale, s, the algorithm starts from Gaussian noise, x which is iteratively denoised while being guided towards the observation.
Algorithm 1 Observation Self-Guidance
Input: observation Vobs. scale x" I) for t = T' to do ---1 ~ (No((x ()) Vx log xx)
xx*
end for
Refinement. The energy-based refinement scheme requires observation, Yobs, base forecaster, g. step size, 7), and noise factor, 7. The pseudo-code the energy-based refinement scheme is shown in Alg. 2. First, Yobs and the prediction of the base forecaster, gyybbs), are combined to ý and used as initial guess, Y(o)- This initial guess is then iteratively refined using the score function of the conditional distribution, po refinement scheme.
Setting ( (91)).
Alg. 2 yields the maximum likelihood
15

---

Algorithm 2 Energy-Based Refinement
Input: observation Yobs, base forecaster g, step size y), noise factor 7, number of iterations N (Yoo)))))))))))))))))))))))
+ Combine obs: (Ybes
(()) y for to . do & 10,
Y(+1) t Y(i) Vy log + 22nii
end for
B Experiment Details
B.1
Datasets
We used eight popular univariate datasets from different domains in our experiments. Preprocessed versions of these datasets are available in GluonTS [2] with associated frequencies (daily or hourly) and prediction lengths. Table shows an overview of the datasets. We used sequence lengths (L. Context Length + Prediction Length) of 360 (i.e., 15 days) for the hourly datasets and 390 (i.c.. approximately 13 months) for the daily datasets. These sequences were generated by slicing the original time series at random timesteps. In the following, we briefly describe the individual datasets.
Solar [29] is a dataset of the photo-voltaic (i.e., solar) power production in the year 2006 from 137 solar power plants in the state of Alabama.
Electricity [29, 11] is dataset of the electricity consumption of 370 customers." 6
Traffic [29, 11] a dataset of the hourly occupancy rates on the San Francisco Bay area freeways from 2015 to 2016.?
Exchange [29] consists of daily exchange rates of eight countries including Australia, British, Canada, Switzerland, China, Japan, New Zealand, and Singapore from 1990 to 2016.
M4-Hourly [37] is the hourly subset from the M4 competition."
KDDCup [16] is a dataset of the air quality indices (AQIs) Beijing and London used in the KDD Cup 2018." 10
ODerTLCCHourll [13] is dataset of Uber pickups from to Jun 2015 obtained from NYC Taxi & Limousine Commission (TLC). !1
Wikipedia [14] is dataset of daily count of the number of hits on 2000 Wikipedia pages.
Table 5: Overview of the henchmark datasets used in experiments.
Dntruct
Chued'S Nome
Thii Stre
Tixet
Demsin
Froq
Median Sog Length
Contect Langh
Prodiction Langh
137
2
7009
Elecmitity
electricity.utt
Tratfic
traffic, nippe
Exrhunge
axchwnge_,rtte sawpnnnge_rttenipeeee
Milifonely
sd_hanrly
KDDCup
kdd, cup, 2018, without, miccing ubex. Flc. hourly
220
270
H
10350
UberTLCCHouryy Hourty
262
262
4520
Wikipedia
w1k:2000__ngg paipe
2000
10000
702
360
$ SSola: http::/www...eeleee/////////dtee "Electricity: http:///wwwo..aaeecocccccccttttt TTraffic: h
/
1KDDDCup: http://wwtrrr.///dddeee////////s
WTherrTlC--ourlyy httpah/hwh....i..oa..oaaaaaaaaaa
16

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B.2 Metrics
Continuous Ranked Probability Score (CRPS). The CRPS is popular metric used to evaluate the quality of probabilistic forecasts. It is a proper scoring rule [15] and is defined as the integral of the pinball loss from 1, i.e.
CRPS(F S. 2Ax w)dx,
where (9. (x-
(vre)) (y 9) the pinball loss for a specific quantile level K. The CRPS measures the compatibility between the predicted inverse cumulative distribution function (also known as the quantile function), F-1, and the observation, y. In practice, the quantile function is not analytically available and the CRPS is approximated using discrete quantile levels derived from samples. We used the implementation in Gluon'TS [2], which defaults nine quantile levels, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), estimated using 100 sample forecasts.
Linear Predictive Score (LPS). The LPS our proposed metric to evaluate the predictive quality of synthetic time series samples from generative model. We define the LPS the test CRPS of linear ridge regression model trained on the synthetic samples. We trained the ridge regression model using 10,000 synthetic samples and computed the CRPS the corresponding real test set. Note that the CRPS of a point forecaster, such as the linear model, is equal to the 0.5-quantile loss (also known as the Normalized Deviation).
B.3 Hyperparameters and Training Details
We trained TSDiff using the Adam optimizer for 1,000 epochs with a learning rate of 1c-3. Every epoch is comprised of 128 batches constructed with 64 sequences each. We used 3 residual layers with 64 channels the backbone of TSDiff. We also added a skip connection from input to output for some datasets as we observed improvements in the validation performance. The number of timesteps in the diffusion process was set to T 100 and we used a linear scheduler with 31 = 0.0001 and 3100 0.1. The diffusion timesteps were encoded into 128- 228ddimensional positional embeddings [53]. Table 6 provides an overview of the hyperparameters.
Table 6: Hyperparameters of TSDiff.
Hyperparameter
Value
Learning
le-3
Optimizer
Adam
Batch size
64
Epochs
1000
Gradient clipping threshold
0.5
Residual layers
3
Residual channels
64
Time Emb. Dim.
128
Diffusion steps T
100
8 scheduler
Linear
0.0001
$100
0.1
Normalization. The magnitudes of the time series can vary significantly, even within the same dataset. Training on raw values of such time series can lead to optimization instabilities. Therefore, we normalized the time series before training. Specifically, we used the mean scaler in GluonTS [2] which divides each time series individually by the mean of the absolute values of its observed context,
norm
Yoba
obs
mean (abs(y
where abs(-) is the element-wing wise absolute function and mean(-) denotes the mean aggregator. After inference, i.c., after computing y norm we rescale the target back to the original magnitude,
Yta yea
meannabssyy......
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Guidance Scale Selection. The guidance scale in observation self- guiddace (see Alg. 1) controls the strength of guidance and was selected on the basis of the validation performance. We tuned the scale from the set {2/32, 3/32, 4/32, 5/32] for mean square guidance and observed that it performs well with the scale 4/32 on every dataset. For quantile guidance, we tuned the scale from the set {1. 2, 4, 8). Table reports the best performing scale for quantile guidance guidance scale also controls the alignment between the diffused time series and the observations. Fig. 5 shows that diffused time series from TSDifI-Q align closely with the observations.
each dataset. The
Ground
Forecast Horizon
Median Prediction
90% Interval
Electricity
150
100
19
20
21
22
23
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01 Sep
200
KDDCup
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1.1
Exchange
11.
Apr
Jul
Oct
Jan
Apr
2012
2013
Figure 5: Predictions generated by TSDiff-Q for time series in Electricity, KDDCup, and Exchange.
Table Guidance scales, s. for quantile self guidance on each dataset.
Dataset
Quantile Guidance Scale
Exchange
Solar
Electricity
Traffic
M4- M4-Hourly
KDDCup
Uber TLC--ourly
Wikipedia
Forecasting with Missing Values. As discussed Sec. 4, we evaluated the forecasting performance of TSDiff under three missing value scenarios to demonstrate its flexibility. Specifically, examined the ability of observation self-gum reff-guidance to handle missing values during inference. For each dataset, we trained a single unconditional model for TSDifT-Q without any modifications to the training objective. During inference, we masked 50% of the timesteps in the context window during inference, e.g., 168 out of 336 timesteps on the Solar dataset. The selection of which timesteps to mask depended on the missing value scenario (see Fig. 6 for an illustration of the three scenarios). Observation self-guidance was then used to predict both the missing and the target values (i.e., the forecast). The CRPS was only computed for the forecast horizon. To ensure that TSDiff did not train on the timesteps that would masked during inference, we excluded window of length equal to (context length prediction length) from the end of the training time series. Consequently, the M4-Hourly and Wikipedia datasets were excluded from the missing value experiments due to the resulting time series being too short.
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Conted
Horeen
Forecast Horion
Ground Truhh
Missing Values
RM
BM-6
BM-
120
150
180
210
240
270
300
330
300
390
Figure 6: Illustration of the three missing value scenarios considered in our experiments.
On the other hand, the conditional models (TSDiff-Cond) were trained individually for each scenario, i.e, the missing values were masked and the model was optimized to predict them correctly.
Computational Cost of Observation Self-Guidance. Table reports the inference time of ob- servation self-guidand against the conditional diffusion model (TSDiff-Cond) on the Exchange dataset while controlling aspects such as batch size, number samples and GPU. The additional overhead for observation self-guidanee (i.e., TSDif-MMS and TSDif--Q) comes from the computation of the gradient neural network (see the Eq. 10).
Table Comparison inference times TSDiff-Cond observation self-guidance on Exchange dataset.
Model Inference Time (seconds)
TSDiff-Cone
162.97
TSDiff-MS
201.08
TSDiff-Q
201.67
Refinement Representative Step. As discussed in Sec. 3.2, we use a representative diffusion step approximate the ELBO instead of randomly sampling multiple diffusion steps. The representative step corresponds to the diffusion step that best approximates the average loss. Fig. shows the loss per diffusion step, the average loss and the representative step, for the eight datasets used in our experiments. We computed the loss per diffusion step on a handomly-samplee batch 1024 datapoints, which took < 13 seconds per dataset single Tesla T4 GPU.
Refinement Iterations Selection. To strike a balance between convergence and computational efficiency, our refinement scheme necessitates careful selection of the number of refinement iterations. Fig.
The relationship between the relative CRPS and the number of iterations is visualized for the Solar dataset. Notably, immediate improvement is observed after just one iteration for point forecasters like Seasonal Naive and Linear. Conversely, DeepAR and Transformer exhibit gradual enhancements. This analysis was conducted solely on the Solar dataset, determining that 20 refinement iterations offer a favorable balance between performance and computational requirements. As result, we the number of refinement iterations 20 for all other datasets and base forecasters.
B.4 Baselines
We compared against eleven baselines, both from statistical literature and deep learning- based models, for the forecasting experiments. In the following, briefly describe the individual baselines.
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un For See
LOI Por 0.%
04
Ofkus Seep
Dikubee Step 05
OPhicen Sime
(a) Solar
(b) Electricity
(c) Traffic
Becrige
0.22
0.30
1.64
02
01
3.00
00
152
(d) Exchange
(e) M4-Hourly
KDDCup
Lifttie
Los ler Piaiio
(g) UberTLC-Hourly
(h) Wikipedia
Figure Loss per diffusion step, L, and representative diffusion step, т. the eight datasets used our experiments.
Seasonal Naive DespAR
Linear
Transformar
Retred CRpS Bazeline CRPS
1.00
0.94
0.92
Refinement Iterations
Figure 8: Variation of the relative CRPS with the number of refinement itcrations on the Solar datasct.
Seasonal Naive a naive forecaster that returns the value from the last season as the prediction, e.g., the value from 24 hours (1 day) ago for time series with hourly frequency.
ARIMA [21] popular statistical model for analyzing and forecasting time series data, combining autoregressive (AR), differencing (I), and moving average (MA) components to capture underlying patterns and fluctuations. We used the implementation available in the forecast package [22] for R.
ETS [21] is forecasting method that uses exponential smoothing to capture trend, season- ality, and error components in the time series. We used the implementation available the forecast package [22] for R.
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TSDIf-Cond Archilecture
Convinc Mr
Conn
Cone
(x),))
Con
N Rosidual Biocks
Swish
( ReLU
Siamoid
T tanh O Elementwwee mut.
Element-wise add.
schematic the architecture of the conditional model, TSDiff Cond. The key difference from TSDiff is the incorporation conditioning input, c, through Convixl layers.
Figure 9:
Linear is linear ridge regression model across time, i.e., it takes the past context length timesteps as inputs and outputs the next prediction length timesteps. We used the imple- mentation available in scikit-leara [40] with the regularization strength of 1 (its default value in scikit-learn) The model was fit using 10,000 randomly sampled sequences with the same context length for each dataset as for TSDiff.
DeepAR [46] is an RNN-based autoregressive model that is conditioned on the history of the time series through lags and other relevant features, e.g., time features such as hour- of-day and sayy-of-weeh The model autoregressively outputs the parameters of the next distribution (e.g., Student's s-t and Gaussian) and is trained to maximize the (conditional) lig-likelihood. We used the implementation available in Gluon [2] with the recommended hyperparameters.
MQ-CNN utilizes convolutional neural network (CNN) architecture capture patterns and dependencies within the time series data. The model is trained using the quantile loss to directly generate multi- horizon forecasts. We used the implementation available in GluonTS [2] with the recommended hyperparameters.
DeepState [42] combines RNNs with linear dynamical systems (LDS). The RNN takes additional covariates (c.g., time features and item ID) input and outputs the (diagonal) noise covariance matrices of the LDS. Other parameters of the LDS such as the transition and emission matrices manually designed to model different time series components such as level, trend, and seasonality. The LDS and RNN are jointly trained via maximum likelihood estimation. We used the implementation available in GluonTS [2] with the recommended hyperparameters.
Transformer is a eequence-to-sequee architecture [53]. The model takes time series lags and covariates as inputs and outputs the parameters of future distributions. We used the implementation available in GluonTS [2] with the recommended hyperparameters.
forecasting model based on the self-attention
TFT [32] is another requence-to-sequeee architecture that includes a variable selection mechanism to minimize the contributions of irrelevant input features. The model is trained minimize the quantile loss at the selected quantiles. We used the implementation available in GluonTS [2] with the recommended hyperparameters.
forecasting model based on the self-attention
CSDI [52] conditional diffusion model for multivariate time series imputation and forecasting. Similar to TSDiff, CSDI is based on the DiffWave architecture [28], but utilizes transformer layers as fundamental building blocks. The original CSDI architecture was presented in the context of multivariate time series comprising temporal and feature transformer layers. We used the original implementation recommended hyperparameters and training setup in our experiments.
the univariate setting with the
TSDiff-Cond is conditional version of TSDiff. It integrates the observed context together with an observation mask using a Conv 1x1 layer followed by an addition after the S4 layer
CSDD implementation: attps://gittae com/ eroongoopp/SrD
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in each residual block as shown in Fig 9. ISDiff-Cond architecture resembles that of SSSD [1] with three key differences:
TSDiff-Cond uses S4 layers in the residual blocks before addition with the conditioning
vector whereas SSSD uses them both before and after addition.
TSDiff-Con is a univariate model where feature dimensions indicate lagged time series values whereas SSSD is multivariate model with feature dimensions indicating the different features in the multidimensional time series.
ISDiff-Conn computes the loss the entire time series, unlike SSSD which only generates the unobserved timesteps. This led to better performance of the conditional model in experiments.
We used the same hyperparameters (e.g.. context lengths and lags) for TSDiff-Cone as for TSDiff (i.e., the unconditional model) to isolate the effect of conditional training.
For the train synthetic-test on real experiments, compared against two popular time series generative models.
Time VAE is based on variational autoencoders and includes network components specific to time series such as trend and seasonality blocks. We used the original implementation and recommended hyperparameters in our experiments.
TimeGAN employs autoencoder to train generative adversarial network in the latent space. The network components are trained using combination supervised, unsupervised. and discriminative loss functions. We used the original implementation" 15 and recommended hyperparameters our experiments.
C Synthetic Samples
Figs. 10 17 illustrate synthetic time series created by VAEE TimeGAN, and TSDiff in comparison to real time series. The generated samples from TSDiff closely resemble the real samples across all datasets, exhibiting higher quality compared to the baselines. For instance, in the case of the solar dataset (Fig. 10), TSDiff accurately captures periods of zero solar energy production during nighttime and generates wide range of peaks within individual time series. In contrast, samples from Time VAE and TimeGAN appear more homogeneous and fail to capture the zero values precisely. TimeGAN also experiences mode collapse issues in certain datasets like Electricity, M4-Hourly, and UberTLCC-Hour while TSDiff consistently generates diverse and realistic samples.
AAAAAAAAAMМИ
MALAA AMM
MAMMMAN
100
200
300
100
200
9000
100
200
300
100
2003
300
Figure 10: Real samples and synthetic samples generated by TSDifT, TimeGAN, and VAE for the Solar dataset.
HTimeVAF implementation: httpa://github... hwwp:/mwaannttttteeee//tttooeeee http://gthhbb.ccmmmaaaeeeaa/////
TimmeGAN implementation: https://github.co http::/gttttttccco///ssss/////a/
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TAM
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100000000000000000
1000000000000000000
200
200
200
300
100
200
Figure 11: Real samples and synthetic samples generated by TSDiff, TimeGAN, and TimeVAF for the Electricity datasct.
TimaGAN
TimelAF
A. M.M.AanANNMm
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Figure 12: Real samples and synthetic samples generated by TSDifT, TimeGAN, and Time VAE for the Traffic dataset.
huy
400
100
400
200
4.00
100
200
Figure 13: Real samples and synthetic samples generated by TSDiff, TimeGAN, and Time VAE for the Exchange dataset.
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Figure 14: Real samples and synthetic samples generated by TSDiff, TimeGAN, and VAB for the M4- Hourly dataset.
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Figure 15: Real samples and synthetic samples generated by TSDiff, TimeGAN, and VAE for the UbeTTLC- Hourly dataset.
Real
TSC#
TimeGAN
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Figure Real samples and synthetic samples generated by TSDiff, TimeGAN, VAM for the KDDCup dataset.
TmeGAN
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Figure 17: Real samples and synthetic samples generated by TSDiff, TimeGAN, and TiE for the Wikipedia datasct.
24


Reviews:
[{"review_id": "sdqJjDKgk8", "reviewer": "Reviewer_PT4A", "summary": "In this paper, the authors proposed an unconditional diffusion model and a self-guidance mechanism for time series data that can be used for conditioning diffusion model for downstream tasks, e.g. time series forecasting and imputation. The effectiveness of proposed model is evaluated from three aspects: prediction, refinement of prediction from other base forecaster, and time series generation. The results show that unconditional diffusion models can achieve comparable forecasting accuracy to conditional ones. The proposed model can be potential useful for other downstream time series analysis tasks other than studied ones.", "questions": "1. The authors described the details of how to do self-guidance without explanations of why this approximation can work in practice. It would be helpful to explain the high level intuition behind it.\n2. The idea of self-guidance seems to be general, can it be used in other domain, e.g. image?\n3. It is a good way to evaluate the model with refining the prediction. But this is not a practical senario. If one has the diffusion model, why not just use it for prediction? There are a lot of spaces used for this in section 4. It would be better for me to discuss other more interesting things.\n4. The authors mentioned that the computational cost of the proposed model is high. It would be better to have such experiments comparing the forecasting speed with conditional diffusion models.\n5. The authors evaluate the time series synthesis using train on synthetic-test on real setup. It would also be interesting to see train on synthetic and real - test on real setup. Can synthetic time series helpful for forecasting?\n6. The experiments are conducted on univariate case, can the method be applied for multi-variate time series?\n7. Can other time series analysis task get benefit from proposed models other than forecasting and imputation? For instance, time series classification, clustering, similarity search.", "limitations": "Yes, the author claimed the limitation of the paper, which is the computational cost.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The author proposed a new self-guidance mechanism to approximate conditional probability p(y_obs|x^t) which enables time series forecasting with unconditional diffusion model\n2. Extensive experiments on multiple benchmarks demonstrate the effectiveness of the proposed unconditional diffusion model comparing to conditional ones.\n3. The paper is well written and easy to follow.", "weaknesses": "1. Refining the prediction of a weak forecaster is a good way to evaluate the model, but it has no real application scenario. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive comments and constructive feedback. In the following, we respond to specific questions raised by the reviewer.\n\n**Comment:** Refinement has no real application scenario / Too much space dedicated to refinement.    \n**Response:** The primary goal of the refinement experiments was to offer an alternative view on the diffusion model. Self-guidance modifies the reverse diffusion process whereas refinement utilizes the implicit probability density of the model. However, we respectfully disagree with the reviewer's comment that refinement lacks practical application scenarios.\n\nRefinement offers a complementary approach that caters to different computational constraints. While self-guidance generally yields superior results, it involves iterating through all $T$ diffusion steps, which can be computationally expensive. In contrast, refinement can be performed for fewer optimization steps, suitable to one's compute budget. Particularly, in the case of simple base forecasters, such as Seasonal Naive and Linear, even a single refinement step yields considerable improvements (see Fig. 5 in the Appendix). This highlights a valuable trade-off between runtime and prediction performance when compared to self-guidance. In industrial forecasting applications, it's not uncommon where one has access to a complex production forecasting system of black-box nature. Refinement provides a computationally efficient yet theoretically sound way to improve forecasts as a post-processor. \n\n**Comment**: What's the high-level intuition behind self-guidance? Why does the approximation work?    \n**Response**: Guidance controls the reverse diffusion process via a conditioning term. Forecasting and imputation involve conditioning on the observed section of the time series. The main intuition behind \"self\"-guidance is that a model designed for complete sequences should reasonably approximate partial sequences. The one-step denoising used in Eq. 6 serves as a cost-effective approximation of the model for the observed time series, providing the requisite conditioning term for guidance in the form of the score function.\n\n**Comment:** Can self-guidance be used in other domains?    \n**Response:** Yes, the idea of self-guidance is general and could potentially be used for other domains and tasks, e.g., images and videos. We are optimistic that future research will adopt our self-guidance mechanism across diverse domains.\n\n**Comment:** Computational cost of inference compared to conditional diffusion models.    \n**Response:** Thank you for your suggestion. We will add a table in the appendix comparing the inference costs of conditional vs. unconditional diffusion models on different datasets. The following table shows the inference time for the conditional and unconditional models on the **Exchange** dataset while controlling for everything (e.g., batch size, number of samples, GPU).\n\n| Model    | Inference Time (seconds) |\n| -------- | -------- |\n| TSDiff-Cond | 162.97     |\n| TSDiff-MS | 201.08     |\n| TSDiff-Q | 201.67     |\n\nThe additional overhead for the unconditional model (i.e., TSDiff-MS and TSDiff-Q) comes from the computation of the gradient of a neural network during self-guidance (see the equation after L128).\n\n**Comment:** Can a _train on real and synthetic_ scenario improve forecasting performance?    \n**Response:** Based on our _train on synthetic-test on real_ results, we are hopeful that data augmentation with synthetic samples would improve downstream forecasting performance. Given other aspects of the current work (self-guidance and refinement), the scope of our generative evaluation was limited to the _train on synthetic-test on real_ scenario. This could be explored in future work. \n\n**Comment:** Can the method be applied to multivariate time series?    \n**Response:** Yes, the ideas presented in this work are not limited to univariate time series. We decided to focus on the univariate case as univariate time series constitute a significant portion of real-world problems. The following are two simple ways of extending the model to handle multivariate time series:\n- Train the model with the channel-independence assumption (popularized by PatchTST [1]), i.e., all feature dimensions of the multivariate time series are treated as independent univariate time series. \n- Modify the backbone denoising network to incorporate a feature embedder for the multivariate time series.\n\nWe will add this discussion to the revision.\n\n**Comment:** Can the proposed models benefit other time series tasks such as classification, clustering and similarity search?    \n**Response:** Yes, we expect the proposed models to benefit other tasks. For example, by augmenting a classifier with synthetic samples or utilizing the imputation capabilities to clean datasets. Furthermore, future work could investigate the possibility of anomaly detection using the implicit density learned by the model.\n\nWe thank the reviewer again for their comments and suggestions. We hope that we have satisfactorily answered the reviewer's questions and clarified sufficiently on the reviewer's main concern regarding the utility of refinement. If so, _we request the reviewer to consider raising their score_.\n\n[1] **Nie, Yuqi, Nam H. Nguyen, Phanwadee Sinthong, and Jayant Kalagnanam**. \"A time series is worth 64 words: Long-term forecasting with transformers.\" arXiv preprint arXiv:2211.14730 (2022)."}, {"review_id": "Hd3S2gcTgK", "reviewer": "Reviewer_D1QU", "summary": "This paper describes a diffusion model for time series problems. Contrary to the popular approach of using a conditional diffusion model, the authors proposed to use the unconditional diffusion model, which is supplemented by a self-guidance mechanism. The authors also proposed a prediction refinement algorithm to improve the prediction of any time series model by using the trained diffusion model.", "questions": "1. In equation (6), as $t$ gets larger, the denominator $\\sqrt{\\overline{\\alpha}_t} \\rightarrow 0$, which is a natural consequence of the diffusion process. How do you handle this singularity issue? In practice, it will not be singular, but potentially cause a numerical instability.\n\n2. The target distribution, $p(y_{ta}|y_{obs})$, but it actually should be $p(y_{ta}|y_{obs}) = p(y_{unobs}|y_{obs})p(y_{obs}|y_{obs})$ and $p(y_{obs}|y_{obs}) = \\delta(y-y_{obs})$. So, upon the guidance denoising, we expect the observed $y$ should be converged to $y_{obs}$. I wonder if the model guarantees this.\n\n3. Provide the proof that equation (9) converges to the maximum likelihood solution when $\\gamma = 0$. Also, please provide a proof that (9) converges to $p(y)$.\n\n4. Carefully reading it, I can understand how the guided diffusion, eqn (5), is computed for missing data and prediction. But, it will be useful if the authors can elaborate it to improve the readability.\n\n5. It is claimed that the model can be used for imputation. Is the diffusion model trained with a missing data? Or, does it require a full data?\n\n*I am willing to upgrade my score once these questions are addressed. ", "limitations": "One of the questions of time series diffusion model is whether it memories the pattern or learns the dynamics. Based on the learning objectives, it is more likely to learn the patterns, which may limit the capability of diffusion-based models for the time series problems of chaotic dynamics and a stochastically forced system. Also, as the authors suggested, the computational cost is certainly of concern for a scalability. But these are beyond the scope of the study. ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Training a conditional diffusion model is challenging due to the high dimensionality of the conditional problem. The authors proposed an interesting method of bypassing training a conditional diffusion model. The proposed methodology is simple and sound.", "weaknesses": "The paper does not elaborate the details of their model, which makes it difficult to fully understand the methodology. For example, it does not clearly show the dimensions of input and output variables, how the missing inputs are treated, and so on.  There are also a few comments in Questions sections. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful to the reviewer for their comprehensive review and appreciation of our work. Our response to specific questions raised by the reviewer follows.\n\n**Comment:** On model details (input and output dimensions, how missing values are treated, etc.).    \n**Response:**\n- The denoising network takes an $L \\times C$ dimensional input, as mentioned in L119, where $L$ is the sequence length and $C-1$ is the number of lags. As is typical in unconditional diffusion models, the output dimension is same as the input dimension.\n- During inference, the missing dimensions are masked when the self-guidance term is computed. Details on missing value experiments can be found in Appendix B3.\n\nWe will further elaborate the model details in the revision. Specific implementation details can also be found in our code which has been released as part of the supplementary material.\n\n**Comment:** How is the singularity of $\\sqrt{\\bar{\\alpha}_t}$ handled in Eq. 6?    \n**Response:** We used a linear beta scheduler with $\\beta_1=0.0001$ and $\\beta_T=0.1$ with $T=100$ in our experiments. This results in $\\sqrt{\\bar{\\alpha}_T}=0.075$ which did not lead to numerical instabilities. However, in our earlier experiments using the cosine scheduler [1], we observed unstable behavior with $\\sqrt{\\bar{\\alpha}_T}=0.0002$. Based on this experience, numerical instability can be avoided by appropriately choosing $\\beta_1$, $\\beta_T$, $T$ and the beta scheduler.\n\n\n**Comment:** Does the observed section of $y$ converge to $y_\\textrm{obs}$?    \n**Response:** Self-guidance enforces a _soft-constraint_ on the observed timesteps. Therefore, the diffused values are not guaranteed to be exactly equal to the observations. The alignment between the predictions and observations for the observed timesteps can be controlled by the scale parameter $s$ (see Eq. 3). In practice, the diffused values for the observed timesteps are close to the observations, as shown in Fig. 1 in the rebuttal PDF.\n\n**Comment:** Proof of convergence of maximum likelihood and Langevin Monte Carlo.    \n**Response:** \n- Note that Eq. 9 with $\\gamma=0$ corresponds to gradient descent which is not guaranteed to converge to the global optimum for general (non-convex) problems. It may converge to a different local optima, depending on the intialization. This behavior is apparent in our results in Table 3 where the scores vary for different base forecasters. We will clarify this in the revision to avoid confusion.\n- For an SDE of the form $dX_t=-\\nabla E(X_t)dt + \\sqrt{2\\gamma}dB_t$, the invariant distribution is the Gibbs-Boltzmann distribution $p(x) \\propto \\exp\\left(-\\frac{E(x)}{\\gamma}\\right)$, where $E$ is the energy function (equal to $-\\log p_\\theta(\\mathbf{y}) + \\lambda\\mathcal{R}(\\mathbf{y},\\tilde{\\mathbf{y}})$ in our case) and $B_t$ is a Brownian motion. This result can be derived using the Fokker-Plank Equation associated with the SDE (see Ch. 4 in [2]). The discretization of this SDE used in Eq. 9, also known as the _unadjusted Langevin algorithm (ULA)_, converges to the invariant distribution given certain regularity conditions (e.g., differentiability and Lipschitz gradients) on $E$ (see [3] for the analysis of ULA). Note that the invariant distribution in our case is not the true underlying distribution $p(\\mathbf{y})$, which is unknown. Instead, we designed the energy function such that low energy is assigned to samples that are likely under the diffusion model $p_\\theta(\\mathbf{y})$ and also close to $\\tilde{\\mathbf{y}} = \\mathrm{combine}(\\mathbf{y}\\_{\\mathrm{obs}},g(\\mathbf{y}\\_{\\mathrm{obs}}))$, ensured by the first and the second terms in the energy function, respectively. We will include this discussion in the revision.\n\n**Comment:** Is the model trained on missing or complete sequences?    \n**Response:** The model was trained on complete sequences for the missing values experiments. The goal of this experiment was to evaluate the model's ability to handle/impute missing values during inference without any knowledge of the missing value patterns during training. We will clarify this in the revision. Further details on the missing values experiments (e.g., data splitting) can be found in Appendix B3.\n\nNote that while the model has been trained on complete sequences in our experiments, it is not a requirement for training TSDiff. Missing values during training can be easily handled by the S4 layers [4] used in our model by appropriately masking the missing timesteps.\n\n**Comment:** Does the model learn the dynamics or the patterns?    \n**Response**: Additional investigation is needed to determine if the model learns the dynamics or patterns. Nevertheless, our results indicate that the model can make reliable predictions for typical time series forecasting datasets and tasks. Future research could focus on evaluating diffusion models' performance with chaotic/stochastically forced time series data, which could yield valuable insights.\n\nWe thank the reviewer again for their insightful comments and valuable suggestions on improving the readability of our work. Based on our responses to the reviewer's questions on the specifics of our model, _we hope that the reviewer would consider raising their score_.\n\n[1] **Nichol, Alexander Quinn, and Prafulla Dhariwal**. \"Improved denoising diffusion probabilistic models.\" In International Conference on Machine Learning, pp. 8162-8171. PMLR, 2021.\n\n[2] **Pavliotis, Grigorios A**. \"Stochastic processes and applications.\" Springer-Verlag New York, 2016.\n\n[3] **Durmus, Alain, and Eric Moulines**. \"Nonasymptotic convergence analysis for the unadjusted Langevin algorithm.\" (2017): 1551-1587.\n\n[4] **Gu, Albert, Karan Goel, and Christopher Ré**. \"Efficiently modeling long sequences with structured state spaces.\" arXiv preprint arXiv:2111.00396 (2021)."}, {"review_id": "Zh5rXCz2LX", "reviewer": "Reviewer_8t5R", "summary": "This paper proposes TSDiff, which is an unconditional diffusion model for time series generation. Besides, the authors propose self-guidance and prediction refinement. The empirical results showcase the superiority of TSDiff over existing baselines on forecasting, refinement, and generating synthetic samples.", "questions": "See Weaknesses.", "limitations": "The limitations are well discussed in the paper.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is joyful to read and the methodology is clearly described.\n2. The empirical results are more than enough. The authors conduct extensive experiments on various of tasks and datasets, making the proposed methodology very convincing.\n3. It is good to see that the authors proposed a new metric to make more reasonable comparison with other baselines in Section 5.3. ", "weaknesses": "1. If y_obs is a subset of timesteps representing the observed timesteps, then is the conditional generation similar to an inpainting problem in 2D images? Then important baselines in solving inverse problems with pre-trained diffusion models, e.g., DDRM [1], should be included, or at least be discussed thoroughly.\n\n2. There are several equations that are not labeled with numbers. It would be the best if the authors can label them. I have several questions regarding the equation after Eq. (10): \n\n[1] Is tau hard to solve? It looks like solving tau is computationally expensive. It would help if the authors can provide some empirical results. Actually I did not quite understand \"tau can be computed efficiently by tracking the running mean for the losses at all diffusion steps during training\".\n\n[2] How does the optimum, tau, vary with different data point? In this equation, the expectation does not take x_t (or y) into account, does this mean we have to solve for tau for every y? Or do we jointly solve a tau using all the data points in the training set?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their review and positive comments on our method and writeup. In the following, we respond to specific concerns and questions raised.\n\n**Comment:** \"...is the conditional generation similar to an inpainting...\" and discussion of image inpainting models.    \n**Response:** At a high level, there are indeed similarities between the time series forecasting/imputation problems we address and the inpainting problem in 2D images. Both involve generating missing or unobserved data conditioned on the observed data. We acknowledge the similarities and cite prior work on inpainting in our related work discussion (see L198). However, there are also key differences. In time series problems, the temporal dynamics and the sequential nature of the data introduce complexities and challenges that are not present in the static 2D inpainting problem. While DDRM provides a powerful framework for linear inverse problems, directly applying it to our problem might not be straightforward due to the aforementioned complexities. That said, we agree that it is an important related work and we will include a discussion on DDRM in the revised manuscript.\n\n\n**Comment:** Missing labels for equations.    \n**Response:** We only added labels for equations that we refer to in the text. We will add labels to all equations in the revision.\n\n**Comment:** Computational complexity of the representative diffusion step, $\\tau$.    \n**Response:** $\\tau$ is not computationally expensive to solve. There are two ways to obtain $\\tau$:\n- **After training**: Compute the average loss over random datapoints for each diffusion step $t$. Then, select the $t$ closest to the average loss over all diffusion steps as the representative step, $\\tau$. This can be done by the following line of code where `losses` is an array of the loss for each diffusion step.\n```python\ntau = ((losses - losses.mean()) ** 2).argmin()\n```\nAs a result, we end up with one $\\tau$ for the entire dataset.\n\nIn our experiments, we computed the loss per diffusion step with a randomly sampled batch of 1024 datapoints. The computation of $\\tau$ took around 13 seconds per dataset on a single Tesla T4 GPU. We will add these details to the revision.\n- **During training**: Rather than computing the losses for each $t$ after the training, we can keep a running average loss for each $t$ and compute the representative timestep using these losses, as above. This is possible because the loss we use to obtain $\\tau$ is the same as the training loss. We will clarify the text in the revision.\n\n**Comment:** \"...the expectation does not take x_t (or y) into account...\" How does the optimum ($\\tau$) vary with different data points?    \n**Response:** Thank you for spotting this typographical error. There should be an expectation over the datapoints as well. We will correct this in the revision.\n\nAs mentioned above, a single diffusion step $\\tau$ is computed per dataset and it does not vary with data points. Fig. 2 in the additional rebuttal PDF shows the loss per diffusion step, the average loss and the resulting representative timestep for the 8 datasets used in our experiments.\n\nWe thank the reviewer again for their review and suggestions. We hope that we have addressed the reviewer's concerns regarding the computational complexity of the representative diffusion step. If so, _we kindly ask the reviewer to consider raising their score_. "}, {"review_id": "LbadBMpHX7", "reviewer": "Reviewer_SUq5", "summary": "The paper focuses on generative diffusion modeling when some of the samples are known, which is advocated as the right setup for regression. Indeed, in a time-series context, those known samples may lie in the past (prediction), or provide boundary conditions (interpolation). \nIn this context, the classical diffusion equations can be worked out to yield a generation that is compatible with the known samples to some controllable extent. This is the key contribution for this paper, and I must say that traming regression as a constrained diffusion is an interesting  idea.\n\nFrom this starting point, the authors try to derive two variants: one where the model estimate is used as a prior, and one where some arbitrary other estimate is used.\n\nA set of forecasting experiments conclude the paper ", "questions": "- L85: consists of decomposing\n- L119: \"see Eq 1\": It is not clear to me why you provide this reference to eq (1)\n- eq (5), again, the choice of a unit variance for the likelihood is weird to me. I would say this is only picked this way to motivate some L2 regularization term in a \"reverse mathemating\" sense\n- I don't understand how you get \\epsilon in equation (10) and further during inference time. \n- L249-250: when you write that the model does not require task-specific training, could you actually explicitly mention *what* is the model trained on ? I thought that there would be a separate training for each task ?\n ", "limitations": "- will you provide some implementation ?", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- the paper is very readable and provides some background and references. It is quite clear\n- Framing regression as a constrained diffusion is definitely interesting and has some merits.\n- I would also say that I'm sure that the proposed method yields interesting results, compared to some other methods", "weaknesses": "Notwithstanding its qualities, I have several concerns with the paper that prevent me from giving it a high score.\n- I would say that the attempt at formulating the method in a Bayesian way is actually a bit awkward in my opinion, because it does not bring much more food for thought than just plainly presenting the guidance as a regularization over the (enegry-based) generating process. Indeed, the prior distribution that is picked in equation (5) just reads like picking some L2 regularization to me, and the quantile regularization that is advocated as working very well in practice just doesn't lend itself very well to a convenient Bayesian treatment. Actually, section 3.2 precisely drops this Bayesian vision to adopt an optimization-based point of view, which seems more natural to me.\n\n- I feel that there is actually a lot of redundancy between 3.1 and 3.2. As I see it, the two \"variants\" are actually the same, with the only difference that for self-guidance, the estimate is derived from the model while it is assumed arbitrary in 3.2.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their review and positive comments on the writing and the merits of our ideas. In the following, we respond to specific questions and concerns raised by the reviewer.\n\n**Comment:** On the Bayesian view of self-guidance and Eq. 5.    \n**Response:** We introduced self-guidance using the Bayes' rule to retain the probabilistic perspective of the guidance term, and to keep the story and notation consistent with the original work on classifier guidance [1]. In our opinion, the Bayesian view is a cleaner and less ad-hoc way to set up the method. We later clarify in the paper that the selected distributions result in the MSE and quantile losses (see L138 and L148). We dropped the Bayesian view in section 3.2 because the _data-space_ refinement scheme makes it confusing to set it up. \n\n**Comment:** Redundancy between 3.1 (self-guidance) and 3.2 (refinement). \"... the two \"variants\" are actually the same, with the only difference that for self-guidance, the estimate is derived from the model while it is assumed arbitrary in 3.2.\"    \n**Response:** We believe that this is not an accurate characterization of the differences between our self-guidance and refinement schemes. The diffusion model estimate is used in *both* schemes. What differs is the interpretation and the sampling procedure.\n- **Self-guidance** modifies the reverse diffusion process with a self-conditioning score function. Ancestral sampling is performed starting from noise at step $T$ all the way to the observation space at step $0$ by iteratively denoising and guiding the sample.\n- **Refinement** interprets the implicit density of the diffusion model as a regularized energy landscape. Rather than starting from noise, it starts from the prediction of a base model and uses Langevin dynamics to sample _directly in the data space_.\n\nThe self-guidance and refinement schemes can also be viewed as the predictor and corrector schemes used in some score-based diffusion models [2].\n\n\n**Comment:** Why refer to Eq. 1 in L119?    \n**Response:** The neural network $\\epsilon_\\theta$ is trained on noisy time series $\\mathbf{x}^t\\in\\mathbb{R}^{L\\times C}$. These noisy time series are obtained using Eq. 1, hence the reference to it.\n\n**Comment:** How to get $\\epsilon$ in Eq. 10?    \n**Response:** $\\epsilon$ is drawn from the standard Gaussian distribution, wheras $\\epsilon_\\theta$ is the denoising neural network (see Fig. 2). $\\mathbf{x}^t$ is the noisy transformation of $\\mathbf{y}$ obtained using Eq. 1. As mentioned in L187, Eq. 10 is a simplification of the ELBO and serves as an approximation of $\\log p_\\theta(\\mathbf{y})$.\n\n**Comment:** \"What is the model trained on? I thought that there would be a separate training for each task?\"    \n**Response:** The model is trained as an unconditional generative model for each dataset. This means that the model is trained to denoise the complete sequence, $\\mathbf{x}^t$. In the *forecasting with missing values* experiment, we trained one model per dataset and used the same model for inference on the different missing value tasks (e.g., random missing, blackout missing). This is in contrast to the conditional models that are trained for specific missing value tasks. Please note that the term \"task\" in our context does not refer to the one used in meta-learning and foundation models literature, as clarified in footnote 1.\n\n**Comment:** Will the implementation be available?    \n**Response:** The implementation has been provided as supplementary material and will be made publicly available at a later stage.\n\nWe thank the reviewer again for their feedback and hope that we have satisfactorily addressed their concerns regarding the Bayesian view and the difference between self-guidance and refinement. If so, _we hope that the reviewer would consider raising their score_.\n\n[1] **Dhariwal, Prafulla, and Alexander Nichol**. \"Diffusion models beat gans on image synthesis.\" Advances in neural information processing systems 34 (2021): 8780-8794.\n\n[2] **Song, Yang, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole**. \"Score-based generative modeling through stochastic differential equations.\" arXiv preprint arXiv:2011.13456 (2020)."}]

Author rebuttal:
We thank the reviewers for their insightful reviews and their constructive feedback to improve the quality of our paper. We are pleased to note that the reviewers appreciate:

- the **technical significance of our self-guidance approach** ("an interesting method of bypassing training a conditional diffusion model", "constrained diffusion is definitely interesting and has some merits", rated *good* on **Soundness** and **Contribution** by 3/4 reviewers);
- the **clarity of our manuscript** ("paper is joyful to read and the methodology is clearly described", "the paper is very readable and provides some background and references. It is quite clear", "paper is well written and easy to follow");
- and, the **thoroughness of our empirical evaluation** ("Extensive experiments on multiple benchmarks demonstrate the effectiveness of the proposed unconditional diffusion model...", "The empirical results are more than enough ... making the proposed methodology very convincing", "I'm sure that the proposed method yields interesting results, compared to some other methods").

Detailed responses to individual reviewers are available under each review. In this general response, we seek to reemphasize the key contributions of our work.

- We present a fresh perspective on time series forecasting and imputation via *unconditional* diffusion modeling, in contrast to prior work that focuses on conditional models. To this end, we propose two novel inference schemes to utilize unconditional diffusion models for conditional tasks during inference.
    - **Observation self-guidance** conditions reverse diffusion for arbitrary forecasting tasks via a guidance term derived from the model's own estimate of the observed time series. The idea behind self-guidance is general and may be applicable to other domains.
    - **Refinement** uses the implicit density learned by the diffusion model to improve forecasts from base forecasters by sampling from an energy-based model.
- We show, through extensive benchmarks, that the proposed self-guidance approach is competitive against task-specific conditional models across datasets and forecasting scenarios. Furthermore, our refinement scheme is able to improve forecasts from base forecasters (especially the simple ones such as Linear and Seasonal Naive).
- We demonstrate empirically that downstream models trained solely on the synthetic samples from the diffusion models generate forecasts of high quality, outperforming existing time series generative models based on VAE and GAN frameworks. This opens up potential research directions analyzing the utility of the synthetic samples for downstream tasks either theoretically or empirically.
- We propose a metric to evaluate the quality of synthetic samples based on a simple Linear regression model which is not sensitive to architecture choices and random initializations, unlike existing predictive metrics.

Meta-review: This paper proposes a diffusion model for time-series forecasting. Different from standard diffusion models, a self-guided mechanism is introduced to align the generation with observations. The model can perform prediction, refinement and synthesis.

Most reviewers think the method is interesting, They raised a number of concerns, which are well addressed in the rebuttal, and some reviewers raised their scores correspondingly. There is a one borderline reject, where the reviewer raised some questions such as writing issues. I believe these issues have been addressed in the rebuttal as well. Thus, I recommend acceptance.
Decision: Accept (poster)