Conditionally Whitened Generative Models for Probabilistic Time Series Forecasting
Authors: Yanfeng Yang
Siwei Chen
Pingping Hu
Zhaotong Shen
Yingjie Zhang
Zhuoran Sun
Shuai Li
Ziqi Chen
Kenji Fukumizu
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Model
Probabilistic Time Series Forecasting
Conditional Generation
Primary area: generative models
TL;DR: A novel class of generative models for probabilistic time series forecasting.

Abstract: Probabilistic forecasting of multivariate time series is challenging due to non-stationarity, inter-variable dependencies, and distribution shifts. While recent diffusion and flow matching models have shown promise, they often ignore informative priors such as conditional means and covariances. In this work, we propose Conditionally Whitened Generative Models (CW-Gen), a framework that incorporates prior information through conditional whitening. Theoretically, we establish sufficient conditions under which replacing the traditional terminal distribution of diffusion models, namely the standard multivariate normal, with a multivariate normal distribution parameterized by estimators of the conditional mean and covariance improves sample quality. Guided by this analysis, we design a novel Joint Mean-Covariance Estimator (JMCE) that simultaneously learns the conditional mean and sliding-window covariance. Building on JMCE, we introduce Conditionally Whitened Diffusion Models (CW-Diff) and extend them to Conditionally Whitened Flow Matching (CW-Flow). Experiments on five real-world datasets with six state-of-the-art generative models demonstrate that CW-Gen consistently enhances predictive performance, capturing non-stationary dynamics and inter-variable correlations more effectively than prior-free approaches. Empirical results further demonstrate that CW-Gen can effectively mitigate the effects of distribution shift.

Published as a conference paper at ICLR 2026
CONDITIONALLY WHITENED GENERATIVE MODELS FOR PROBABILISTIC TIME SERIES FORECASTING
Yanfeng Yang"
Graduate University of Advanced Studies & The Institute of Statistical Mathematics Tokyo, Japan
yanfengyang 60 il 1cco
Siwei Chen, Pingping Hu, Zhaotong Shen, Yingjie Zhang, Zhuoran Sun, Shuai Li, Ziqi Chen
East China Normal University,
Shanghai, China
zqchen0 fem. ecnu. edu. on
Kenji Fukumizu"
The Institute of Statistical Mathematics Tokyo, Japan
fukumi zuê sm. a jp
ABSTRACT
Probabilistic forecasting of multivariate time series is challenging due non- stationarity, inter-variable dependencies, and distribution shifts. While recent dif- fusion and flow matching models have shown promise, they often ignore informa- tive priors such as conditional means and covariances. In this work, we propose Conditionally Whitened Generative Models (CW-Gen), framework that incorpo- rates prior information through conditional whitening. Theoretically, we establish sufficient conditions under which replacing the traditional terminal distribution of diffusion models, namely the standard multivariate normal, with multivari- ate normal distribution parameterized by estimators the conditional mean and covariance improves sample quality. Guided by this analysis, design a novel Joint Mean-Con -ovaarCCve Estimator (JMCE) that simultancously learns the condi- tional mean and sliding-window covariance. Building JMCE, introduce Conditionally Whitened Diffusion Models (CW-Diff) and extend them to Con- ditionally Whitened Flow Matching (CW-Flow). Experiments on five real-world datasets with six state-of-the-art generative models demonstrate that CW-Gen con- sistently enhances predictive performance, capturing non-stationn stationary dynamics and inter- variable correlations more effectively than prior-fre approaches. Empirical results further demonstrate that CW- can effectively mitigate the effects distribution shift.
INTRODUCTION
Time series analysis has a long history, with classical approaches such as ARIMA, state-space models, and vector autoregressions (VAR) (Box & Jenkins [976; Durbin Koopman, 2012 Lütkepohl 2007). Although these methods have been widely applied, they often struggle with high-dimensionality and complex data structures that arise in modern applications. More recently, neural architectures have demonstrated superior predictive accuracy, such recurrent neural net- works (RNN), Long Short-Term Memory (LSTM), and Transformers (Sherstinsky) [0200] Hochre- iter & Schmidhuber 1997 Vaswani et al. 2017). However, these neural models primarily focus forecasting the conditional mean of future sequences given historical observations, while providing
All authors contributed equally. "Corresponding authors.

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little to uncertainty quantification. These limitations have motivated the development of probabilis- tie forecasting, which seeks to model not only point predictions but also the associated uncertainty.
Multivariate time series probabillistic forecasting has recently emerged as a key methodology for quantifying predictive uncertainty, enabling informed recision-making in numerous real-world ap- plications in diverse domains such as finance, healthcare, environmental science, and transportation 2021). Formally, the task involves learning the probability distribution Pxyc of a future time series Xo E [RdTT of discrete time conditioned on its corresponding historical obser- vations C E RRxxTh, where the integers Ty and T₁, denote the lengths of future and historical time series, respectively, and d represents the dimensionality of each time step. However, this task still remains highly challenging, primarily due to (i) non non-stationary characteristics, manifested through long-term trends, seasonal effects, and heteroscedasticity (Li et al. [024: Ye et al. 2025); (ii) com- plex vatrivar dependency structures (Yuan & Qiao 2024); (ii) inherent data uncertainty, such as short-term fluctuations (Ye et al.] )0225; and (iv) potential distribution shifts between training and
(Lim & Zohren
testing data (Kim et al. 2022).
In response to these challenges, recent advances in generative learning, especially diffusion mod- els, focus on accurately estimating the conditional distribution Pxyc- TimeGrad employs a RNN to encode historical observations and generates forecasts autoregressively, but suffers from cumulative errors and slow computation (Rasul et al. 2021). CSDI uses a 2D-Transformer for imputation and forecasting (Tashiro al.] 2021). while SSSD employs Structured State Space Model reduce computational cost and emphasize temporal dependence (Alcaraz & Strodthoff, 2023). Neverthe- less, CSDI, SSSD, and TimeGrad all struggle with long-term forecasting (Shen & Kwok 2023). Diffusion-TS leverages transformer to decompose time series into trend, seasonal, and residual components for generation, whereas FlowT'S accelerates generation using rectified flow (Yuan & Hu ct al. 2025).
Qiao 2024
Although the aforementioned generative models have achieved promising performance, they ignore informative priors. Such priors, derived from historical observations or auxiliary models, can sub- stantially improve conditional generative modeling. To the best of our knowledge, CARD is the first model incorporate prior information into conditional diffusion models (Han et al. 2022). It pretrains a regressor to estimate the conditional mean E [X,[C] [C] and integrates this regressor into the diffusion process, thereby enhancing conditional generation. In time series forecasting, regressing the conditional mean and incorporating it into diffusion models as a prior has become a common practice, as it alleviates the difficulty of modeling non-stationary distributions. Time adopts a linear regressor to capture short-term patterns and employs future mixup strategy during training to mitigate boundary disharmony (Shen & Kwok 2023). However, its linear design limits the ability to capture complex trends and fluctuations. TMDM addresses this limitation by integrating a non- linear regressor into the variational inference framework, enabling joint training of the regressor and the diffusion model (Li et al. 2024). The regressor for E [Xo] CJ (hereafter referred to as the mean regressor) can capture trends, seasonality, and fluctuations but vulnerable to heteroscedasticity. Building on this line, NsDiff addresses this by introducing two pretrained models: and a variance regressor, the latter estimating the conditional variance of each variable within a slid- ing window (Ye et al. 2025). By incorporating both regressors into the diffusion process, NsDiff models heteroscedasticity more effectively. Despite these innovations, the method still suffers from certain limitations, particularly the overly complex reverse process and the neglect of correlations among variables. A detailed discussion of these limitations is provided in Appendix A Beyond diffusion models, S2DBM employs a diffusion bridge variant and incorporates the mean regressor in the same manner as CARD Yangg et al. 2024). which limits its ability to handle heteroscedasticity. TsFlow uses Gaussian Processes (GPs) both the mean and variance regressors (Kollovich et al.) 2025), but its design is restricted to univariate forecasting with short horizons and inherits the typical drawbacks of GPs, including kernel sensitivity and cubic computational cost.
regressor
Building on the preceding literature, it is well established that carefully designed priors can sub- stantially enhance generative models. Yet several key questions remain unresolved: How exactly do priors contribute to these improvements, and how accurate must the mean and variance regressors be to provide tangible benefits? How can such regressors be effectively trained, and are there theo- retical guarantees supporting their impact? Most existing approaches incorporate mean and variance regressors into diffusion models by following the designs of CARD and DDPM (Han et al. 2022
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Ho al. 2020). This raises further question: is this mechanism redundant inefficient, and could it be simplified within more flexible diffusion frameworks?
Motivated by these questions, introduce the Conditional Whitened Generative Models (CW- Gen). Our main contributions are:
We develop unified framework for conditional generation, CW-Gen, with two instantiations: the Conditional Whitened Diffusion Model (CW-Diff) and the Conditional Whitened Flow
Matching (CW-Flow). Several prior methods (Han et al. 2022 Liet al. 2024 Ye et al. 2025 can
be viewed as special cases this framework. Furthermore, CW-Gen allows seamless integration with diverse diffusion models.
We provide theoretical analysis that establishes sufficient conditions under which CW-Gen im- proves sample quality, as stated in Theorem and Theorem Q Appendix
Motivated by Theorems [ and 2 we propose novel joint estimation procedure for the condi- tional mean and sliding-window window covariance of time series. Empirically, it achieves high accuracy while effectively controlling covariance eigenvalues, ensuring stability and robustness in genera- tive modeling.
We integrate CW-Gen with six state-of-the-arr generative models and evaluate them on five real- world datasets. Empirical results show consistent improvements in capturing non-stationarity, inter-variable dependencies, and overall sample quality, while also mitigating distribution shift.
PRELIMINARIES
2.1
DENOISING DIFFUSION PROBABILISTIC MODELS (DDPM)
Most of the diffusion models discussed in Section follow the DDPM framework (Ho et al. 2020) which we review below in general conditional setting. Let (Xp. C) be random vector with the joint distribution Px.c, where Xo Rd and C' E R The (conditional) DDPM aims to learn the conditional distribution Pxic and generate samples that match this distribution through forward and reverse process. In the forward process, Gaussian noises gradually added into Xo by stochastic differential equation (SDE):
dX,- 8, VB,dW,, TE [0, Xo Pxic,
where B, > O and W, is Brownian motion in Rd=,, We use for the time of diffusion throughout this paper, while is the index for time series. From the propertics of Chstein-Uhleeneeck-eeee (OU-process), derive the marginal distribution of X,:
X, - a, JyC, E ((,,
where or exp {- Bads/2), = = oz denotes equality in distribution, and la, the dy di-dimensional identity matrix. By construction of 3,, the integral fo 3,ds is sufficiently large, so the distribution of Xj (the terminal distribution) is well-approximated by (0, Ia.). In the reverse process, standard Gaussian noise X, is gradually denoised by an SDE:
dx , , Br v log (X+ + VB,dW,,
Tmins with Tmin being early stopping time close to O, and W, , is Brownian motion. In Pr (-CC) and V log (-(CC) denote the conditional density and score function of X, given C. respectively. Since the conditional score function is intractable, [Ho al. (2020) and al. (202)) proposed approximating it with neural network se parameterized by 0, trained by minimizing:
where starts from 1 and ends
E E(xo,C),7 ||s@ (a+Xu + o.c, +,o)+
where (0, 1] and N(O, Ia.). Finally, substituting Vz log (X+IC) with Sol C,T) yields the reverse process:
dx,- [- 1,,x, 3r80 C,,,,,,))) C, + VBraW, v. [7min 1).
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2.2
FLOW MATCHING
Unlike diffusion models based on SDEs, Flow Matching (FM) employs an ordinary differential equation (ODE) connect Gaussian noise N(0, Ia_) with the data Xo Pxic (Lipman al. 2023
dx, (c- (o)Jo) TE [0, 1].
A neural network Up, parameterized by vb, learns the vector field of 2 by minimizing:
(xx,,C) Xo - vi ( + T(e - Xo), C,
Given the learned vector field, FM generates samples by solving the ODE:
dX, -x C,T)dT
to Tmin where + 1 is Gaussian noise. The final state X is the generated sample.
from
THEORY AND JOINT MEAN-COVARIANCE ESTIMATOR (JMCE)
3.1 THEORETICAL FOUNDATION
key question addressed in this subsection is how modifying the terminal distribution ((,, Ia,) enhance generation quality. The total variation distance between the generated distribution of a diffusion model and the true distribution grows as the convergence error of the forward process increases, where the latter involves the Kullback-Leibler divergence (KLD) between Pxic and the terminal distribution DKL (xx || (O, Ia,)) as factor in the error (Oko al. 2023 Chen al, 2023 Fu et al. 2024). Hence, smaller value of this KLD leads to samples that better match Pxic- This insight motivates replacing the standard terminal distribution N(0, Ia.) with N(uxcc: Ex(e), where μ and Σxie are the true conditional mean and covariance of X given C. Since these
quantities are unknown in practice, they must be estimated by ixjc and Exic. The advantage of this replacement can then measured by the reduction in
DKL Pxic |l [[xxc (x(x))xxx(c) relative to DKL (Px N(O, Id.)).
This raises the fundamental question of when replacing the terminal distribution N(0,, Ia,) with [(xx c, Exic) improves generation quality, which the following theorem addresses.
Theorem Let Pxic denote the true conditional distribution of X E Rd given C, with conditional mean Pxic and positive-defintte conditional covariance E Define Qo = N(0, Ia.) and Q == N c. Exio). where fxicc and Exic are estimators of px and Σx c, respectively. Let Xxic, denote the i-th eigenvalues of Exic, for i = 1,2....,, A sufficient condition ensuring that (PI @) < (KI(P Q0) is:
min AxICs) XxIC)))((IAEE)))) (IR2ICс- -1Ixxc-- +xicll++ [E Expelln)
(3)
+ +Vde |Eic - ExiollE I 1xxicl...
where Exic - Sxcl1 - 8s and 8y is the i-th singular value of Exic Ex
Theorem states that when (3 holds, replacing Qo with Q reduces the KLD between Pxic and the terminal distribution, thereby improving generation quality. Importantly. it provides foundation for designing loss functions to estimate иI|с and Exic. as detailed in Equation (4) below. We emphasize that the estimators of μ. C and Σxie are obtained by minimizing the sample counterpart of the left- hand side of (3) as detailed in the next subsection.
In order for 33 to hold, it is necessary to obtain accurate estimators of both P'|c and Σxic- The estimation accuracy of Σxic measured in terms both the Frobenius norm and the nu- clear norm, with the latter characterized by Ede 8t. We employ Cholesky decomposition
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and introduce penalty term into the loss function to enforce that the smallest eigenvalue, remains strictly positive and bounded away from zero, as detailed in the next subsection. Furthermore, in non-stationary time series, И often exhibits sharp variations and thus deviates from zero. Consequently, (3) is more likely to hold when accurate estimators of both |x|c and Ex(c are available. Conversely, (3) may fail to hold in unfavorable regimes-for example, when the signal magnitude x 12 is small, the estimators of M and Ex are in- accurate, or the inverse of the smallest eigenvalue becomes large. In such cases, incorporating the corresponding prior models can potentially degrade performance. In the next subsection, design a novel loss function to mitigate this risk. A detailed discussion can be found in Appendix D
d. {Axicaa) (l.../
We further identify the scenarios in which our proposed replacement outperforms TMDM and Ns-
Diff (Li et al. 2024: Ye et al. 2025), as formally established in Theorem 2 in Appendix C
3.2 JOINT COVARINCEE ESTIMATOR (JMCE)
Theorem D establishes that accurate estimators of both the conditional mean and covariance can improve the quality of samples generated by diffusion models. Guided by the sufficient conditions we design novel Joint Mean -ovaariiee Estimator (JMCE).
In terms of time series, directly estimating the true conditional covariance is extremely challenging, as it often highly complex and non-smooth, which makes consistent estimation difficult. Instead, the sliding-window window covariance is preferable, as it not only offers more accurate approximations but also improves computational efficiency (Iwakura al. 2008 Chen et al., 2024). Motivated by this, estimate the Hiddng--innoww conditional covariance, rather than the true conditional covariance.
Let Exo.t Rdxd denote the sliding- window covariance time I, and let E X Rdxd be an
estimator of Exo.t for = 1,...,Ty TG. We design non-autoregressive model to simultaneously output:
Cxic: Lijc.... LT,IC- JMCE(C)
with Σxp,1Ic == EnjcITe- for 1, ,, and Lne are lower-triangle matrices. This de- sign, inspired by Cholesky decomposition, guarantees that E.x., are positive semi-definite (PSD). The detailed algorithm of JMCF(C) can be found in Appendix B In our implementa- tion, we use a Non-stationary Transformer (Liu et al. 2022) as the backbone of JMCE. Based on (3) in Theorem [
construct the trainning loss in JMCE by combining three compo- LF == Exx..c) EE, [Ex., Ex., and LSVD =
nents: L2 = E(x..c) Xo Pxcellz
C) Σ Exo. - Exx.1eely The smallest eigenvalues of Ex,,ti have crucial impact on the magnitude of the left-hand side of inequality
We thus introduce regularization term that enforces the smallest eigenvalues of Ex..,IC to remain strictly positive and bounded away from zero, thereby avoiding numerical instability and rank deficiency. Let Amin be tunable hyperparameter. The penalty term is defined as:
Rumn Exp,1cc
ReLU (Amin - sinid))
where Axxx, 1....,d) d) denote the eigenvalues of Σx, c, and ReLU(x)) = max(x, 0}. It is indicated that any eigenvalue smaller than Amin will be penalized. The overall training loss JMCE for the conditional mean and covariance is defined as:
T Σ RAm (Ix,.eic).
LIMCE = L2 + LSVD + Amin / TT-TFFFF F + WFigen
(4)
where WEigen is a hyperparameter that controls the strength of the penalty. Empirically, we choose WHigen ~ -O--1 It important to note that (4) is specifically designed to ensure that (3) holds.
The algorithm of the joint estimator can be found in Appendix B JMCE excels at estimating the conditional mean and covariance while controlling the minimal eigenvalue. We conduct a substantial ablation study to show the advantages, and discuss them in Appendix E
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4 CONDITIONAL WHITENED GENERATIVE MODELS (CW-GEN)
In this section, we propose Conditionally whitened diffusion models (CW-Diff) and Conditionally whitened flow matching (CW-Flow). Together, we call them Conditionally Whitened Generative Models (CW-Gen).
4.1 CONDITIONALIY WHITENED DIFFUSION MODELS (CW-DIFF)
Our JMCE outputs c E Rd and Ex,Ic == [Exo. E xxo,T,icl E RdxdxTy, Since all Ex are positive definite, we can compute Etk 2x,CC = [S 1........... Stk 2,cc RaxdxTyy for RdxT,, where each column ~ (0, Ia) and the columns (1....,(2. Cry are mutually independent. We define the tensor operation
k {-0.5, 0.5) via eigen- ggen-decompooition. Let E == (EL,...,
00.5 C, 06 :6:==
(50.5 1|, -
55..5 61,-..,
Xx,Ty/C Xx, eT,l E BdxTT
Accordingly, we say that a tensor follows N N c. Ex,ic) if it has the same distribution as -x, + Nxjcc With this formulation, we define the forward process:
50.5
d d(x, - (xxcc - 3 (x+ PxIc)dT + Br
$0.5 2x/e dW,, T (0, Xo Pxc.
(6)
where W- is a Brownian motion in RRdxT, By the property of the OU-process, the terminal dis- tribution of X1 is close to N c, Ex,ic.. .) A formal proof of the terminal distribution of (6) is provided in Appendix(C Furthermore, the following SDE is equivalent to (6): d S-0.5 "xa(C" o (X, - c = - 3, 5-0.5 X.IC
(X, Ti dr + VBrdW, T [0 [0,
which implies that the diffusion processes can be directly performed on X. = S xx
5-0.5 (Xo -x(c)-
We call this operation conditional whitening (CW). Subtracting c removes the non-stationary trends and seasonal effects in Xo. while being operated by "xx/CC addresses heteroscedasticity and mitigates linear correlations among features. The CW operation thus renders the data stationary as possible and enables diffusion models to more effectively capture temporal and higher-order depen- dencies. Moreover, since it full-rank linear transformation, CW is entirely invertible. Building on these properties, we now formally write the forward process of the Conditional Whitened Diffu- sion Model (CW-Diff) follows:
dx = 3 CW dt + d, T E [0, 10,
with the initial state XCW satisfying (S2.jc Xow fxic) Pxic- Correspondingly, we use neural network sg to learn the score function of x given C by minimizing the following loss function:
Exp sl 1 (a,X0 o,e, C, e/o+/1
Let X ~ NO, laxaxt,), where laxaxty == [lay..., la] E Rdxxxxfy Then, the reverse process of CW-Diff is given by:
dx - CWw 3, (Xew C,r)) dr + W.dW
where 7 decreases from obtain
Tmins with Tiin being an early stopping time close 0. Finally, we
50.5
CalC
C Trie
Axic
by inverting the original CW operation. ing Cx
Trin is the final sample generated by CW-Diff approxxiat
The forward process in Equation is consistent with that of DDPM. Furthermore, CW-Diff is readily extendable TMDM, NsDiff, and other diffusion models. This extension is accomplished by replacing the initial variable X, with its CW-transformed form xCW Within this framework, dlding-window the task of learning the mean and sliding window covariance in x may be interpreted form of residual learning, analogous to the mechanisms used in GBDT and XGBoost (Chen & Guestrin 2016).

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Conditionally Whitened Diftusion Model (CW-Diff)
Original
Conditiooally
Train penetate samples Reverse coeditiondl wintening
4g
Conditionally Whitened Flow Matching (CW-Flow)
Train JMCE
Mean: Covariacee Isttmmtto
Generate white
Convert white
UWiFlow and
Figure 1: The flow chat of JMCE, CW-Diff and CW-Flow.
4.2 CONDITIONALLY WHITENED FLOW MATCHING (CW-FLOW)
In CW-Diff, the inverse matrices of Σ xx, computed via eigen-decompositton, which requires computational complexity of O(dT)). To reduce this cost and improve efficiency, we transition to the FM framework introduced in Section 2.2 where the estimated mean and covariance can be
incorporated in more efficient way.
The Conditional Whitened Flow Matching (CW-Flow) model employs an ODE to connect Xp
Pxic with noise cW ([xic Ex, c):
dx -ce (Xo)dr) T [0, 1].
Accordingly, the CW-Flow network o is trained by minimizing:
,,,, ||e - Xo Cx (Xo T(E - Xo)) C, c,
CW-Flow then generates samples by solving the following ODE:
dxcw -
The algorithms of CW-Diff and CW CW T)d,, XCW ~ Ex,1c,
where 7 starts from and ends at T Tmin- XCW is the final sample generated by CW- Flow approximating Pxic. Compared with CW-Diff, CW-Flow does not require computing inverse
Tmin
matrices reversing the CW operation of the final sample X
Train CW-Flow are provided in Appendix B The flow chart of CW-Diff and CW-Flow can be found in Figure
EXPERIMENTS
Datasets: We evaluate CW-Gen five representative time series datasets-ETTh,, ETTh2, ILI, Weather, and Solar Energy-spanning various domains and temporal resolutions. Further details of the datasets can be found Appendix [..T For the ETT datasets, the training/validationnness split follows 3:1:1 ratio, while for the other datasets we adopt a 7:1:2 ratio. Table [ presents the dataset properties and the win rate CW-Gen, computed as the proportion of cases where CW-Gen outperforms competing methods, based on the results in Tables
Baselines: We evaluate five diffusion models and one flow matching model for time series forecast- ing (denoted as Raw), and further integrate all six generative models with our CW-Diff and CW-Flow approaches (denoted as CW). Specifically, the baselines include TimeDiff (Shen & Kwok
2023
SSSD (Alcaraz & Strodthoff 2023). Diffusion-TS (Yuan & Qiao 2024), TMDM (Li et al.
2024
NsDiff (Li et al. (2024), and FlowTS (Hu et al. 2025). Among them, Diff, TMDM, and NsDiff prior-informed methods.

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Metrics: We evaluate the predictive performance with six metries: Continuous Ranked Probability Score (CRPS) (Matheson & Winkler 1976). Quantile Interval Coverage Error (QICE) (Han et al, 2022). Probabilistic Correlation score (ProbCorr), Conditional Context Fréchet Inception Distance (Conditional FID) (Yue et al. 2022), Probabilistic mean square error (ProbMSE), and Probabilistic mean average error (ProbMAE), Formal definitions be found in Appendix F.2
We also provide
the results for ProbMSE and ProbMAE in Tables 7 and 8 in Appendix E.3
Settings:
During evaluation, Xo and C refers to non-overlapping subsequences drawn from the test set, where C denotes the historical observations and Xn the corresponding future series. We adopt the widely used long-term forecasting setting with a historical length of 168 and a future horizon of 192 (Shen & Kwok [023] Ye et al. (025). Inspired by NsDiff, The sliding-wing window covariance computed with a window size of 95, except for III,, where
set to 15 Ye et al., 2025). In the JMCE loss 44), Amin is fixed at 0.1, and the penalty weight Wrigen is set to 50. All diffusion models follow their default diffusion schedules, and the number of sampling steps is set to 50 (20 for NsDiff). We train JMCE and CW-Gen the training set, select the model checkpoint with the lowest loss on validation set, and then perform evaluation on the test set. Each model generates 100 samples for evaluation. On each dataset, train every model 10 times with different random seeds and report the mean and one standard deviation of the four metrics. We also conduct extensive ablation studies on JMCE, which can be found in Appendix E.4 The other parameters are provided in Appendix F
As shown in Tables 6 CW-Gen reduces CRPS and QICE in substantial number of cases, indicating improvements in predictive accuracy. Moreover, it consistently lowers ProbCorr and Conditional FID, with only minor exceptions, showing that CW-Gen enables models to better capture feature correlations in time series and to enhance overall sample quality. Moreover, as shown in Tables and our CW-Gen method improves the ProbMSE metric in 76.67% and the ProbMAE metric in % of the evaluated model dataset combinations. This demonstrates that, in addi- tion to enhancing probabilistic forecasting ability, CW-Gen also strengthens the point forecasting performance of the models.
Results:
Illustrations: In Figure 2 we illustrate representative results of representative generative models combined with CW-Gen. Among them, Diffusion-TS serves typical diffusion model, NsDiff a diffusion based model augmented by priors, and FlowTS is based on flow matching. Compar- ing NsDiff and CW-Gen with the other models, observe that generative models without priors tend to generate sample with shifted means and variances, which we attribute to distribution shifts between the training and test sets. This observation highlights the benefit of incorporating priors in probabilistic time series forecasting, as they can effectively mitigate such distribution shifts. In contrast, "W--Diffusion-TS and CW-Flow'TS, which leverage JMCE as priors, exhibit noticeable mean shift compared to Diffusion-TS and FlowTS. Moreover, the individual samples generated by CW-Diffusion-TS and CW-FlowTS achieve finer resolution and better capture the peaks in Dimen- sion than their non- CW counterparts. Compared with NsDiff, CW-NsDiff produces more accurate sample means and smaller standard deviations in Dimension which contributes to more reliable uncertainty quantification. More illustrations can be found in Figure 3 in Appendix E
Table 1: Dataset descriptions, including dimensions d, frequenciess total length of time series, length of historical observations Ti, length future time series Ty, and win rates our CW methods. Win rate refers the proportion that CW method outperforms original method.
Dataset
Dimension
Frequency
Total length
Th
T
Win rate of CW-Gen
ETTh1 ETTh2
7
Hour
14,400
168
192
22/24
91.67%
7
Hour
14,400
168
192
22/24 =s 91.67%
ILI
7
I Week
966
52
36
20/24 83.33%
Weather
21
10 Minutes
52,696
168
192
22/24 xs 91.67%
Solar Energy
137
10 Minutes
52,560
168
192
19/24 79.17%
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Table 2: Metrics for models trained on original ETThl (Raw) and conditionally whitened ETThl (CW). Each experiment is repeated by 10 times, and standard deviations are provided The better results between Raw and CW are underlined. The win rates of every metric of Raw and CW-Gen models are also provided.
in brackets.
Model
CRPS (L)
QICE ())
ProbCorr (1)
Conditional FID (1)
(ETTh1)
Raw
CW
Raw
CW
Raw
CW
Raw
CW
TifeDiff
0.787 (0.051)
0,505
9.038
8.821
0.320
0.243
19.008
6,788
2023)
(0.040)
(0.946)
(1.916)
(0.012)
(0.027)
(6.088)
(5.425)
SSSD
0.836
0.524
11.624
4.838
0.326
0.238
40.887
9.265
(2023)
(0.153)
(0.085)
(1.312)
(1.921)
(0.032)
(0.024)
(17.601)
(5.003)
Diffusion
0.626
0.445
3,002
2.963
0,401
0.266
81.563
7.686
-TS (2024)
(0.027)
(0.024)
(0.838)
(0.887)
(0.017)
(0.012)
(60.905)
(2.751)
TMDM
0.472
0.440
3.360
4.555
0.230
0.213
9.931
3.831
2024)
(0.031)
(0.001)
(1.055)
(0.855)
(0.014)
(0.001)
(4.439)
(0.431)
NsDiff
0.407
0.431
1.792
1.249
0.214
0.206
35.261
8,820
2025)
(0.032)
(0.029)
(0.682)
(0.228)
(0.014)
(0.010)
(7.785)
(1.541)
FlowTS
0.724
0.488
8,820
8.817
0.354
0.254
39.793
4.865
(2025)
(0.135)
(0.020)
(2.631)
(0.460)
(0.060)
(0.021)
(24.853)
(0.563)
Win rate
16.7%
83.3%
16.7%
83.3%
0.0%
100.0%
0.0%
100.0%
Diffusion-
NsDiff
FlowTS
58
ETThl, Dim
50 13
100 17
TW-Diffusion-TS
CW-NsDiff
CW-FlowTS TS
255
Diffusion- TS
Nsfiff
FlowTS
ETThl, Dim= =
CW-Diffusion-TS
CW-NsDiff
CW-FloWTS
Figure 2: Comparison of Diffusion-TS, NsDiff, FlowTS, and their CW variants ETTh1 across Dimensions and True ETThl means the real time series from ETThl dataset. Sample mean and standrad deviation refer to the mean and standrad deviation of 100 samples generated by generative models. One sample refers to a randomly chosen instance among the 100 generated samples.

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6 CONCLUSION
In this work, we establish for the first time a sufficient condition that reduces the KI. divergence between conditional distribution and the terminal distribution of diffusion model. By tightening this KL divergence, we obtain sharper bound on the total variation distance between the generated distribution of the diffusion model and the true distribution. Building on this result, we design the Joint Mean- Covarianee Estimator (JMCE), which jointly estimates the conditional mean and the conditional Budding-window covariance while controlling the behavior of the minimal eigenvalue. We then use JMCE data-driven prior to conditionally whiten the original data, and train diffusion models on the whitened space, yiclding the Conditionally Whitened Diffusion Model (CW-Diff)), Similarly, by modifying the terminal distribution of flow matching, we introduce the Conditionally Whitened Flow Model (CW-Flow). Together, we refer to these as CW-Gen. We evaluate CW- Gen five real-world time series datasets using six generative models and four evaluation metries. Experimental results demonstrate that CW-Gen consistently improves model performance in most cases.
REPRODUCIBILITY STATEMENT
Our proposed CW-Gen models presented in Section and the corresponding algorithms provided in Section B Theorem can be found in Section 3 with its proof given in Appendix C.2 In addition, we introduce Theorem
Appendix [ and its proof is provided in Appendix C.3
Detailed descriptions of the datasets, models, and evaluation metrics used in our experiments are included Section Appendix and Appendix [ wtwiioonnlllltteeeeeeeeeeeeeeeee
The code is available at: / whi whitened tened www.rativeeeeoeeeeeeeeeee me
com/
0316, Coodit conditioonall
ACKNOWLEDGMENTS
This work has been partially supported by JST CREST JPMJCR2015 and JSPS Grant-in-Aid for Transformative Research Areas (A) 22H05 106. Yanfeng Yang is supported by JST SPRING, Japan Grant Number JPMJSP2104. Dr. Ziqi Chen's work was partially supported by National Natural Science Foundation of China (NSFC) (12271167 and 72331005).
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A RELATED WORKS
A.1 TRANSFORMER SENSFORRMR-MMDDULATEEE DIFFUSION MODELS (TMDM) AND NON-STATIONARY DIFFUSION MODELS (NSDIFF)
Han et al.
2022 incorporated an estimator of the conditional mean into the DDPM framework,
naming this approach CARD. TMDM later adopted this framework for time series forecasting (Li et al. 2024). Recall Cxic E pdxTf is an estimator of Xo given C. The discrete forward process of
TMDM 18:
X [l|
V Blm ( [[n +
B(n)) c + Vingenn n I......,NN
where 0 Cx and 6/n) ~ N(0, IdxdxT,) for all n. N is sufficient large index. By incorporating F into the forward process, the model is able more effectively handle the non- stationary trends and seasonal effects.
To further mitigate heteroscedasticity, Ye al. (2225) proposed NsDiff, which introduces estimators for the sliding-window variance of the time series. Let L t E Rddd denote the diagonal sliding-
variance matrix time f. We then define ox T (o,l::--. o To E RdxdxT; for (1,2)-
We also introduce 22 Xx/C = T 1 .
2 XyIT/ICl E RaxdxTy as an estimator of Rx The
discrete forward processes of NsDiff is:
p V-- Sqml ixjc+ Bin + PPg (orocc - -oicc)o2)) ) 0.5
[nn] - / [n
oE (8)
In the reverse process, xx is unknown; NsDiff estimates by exploiting both [[n] and @2 x, rather than relying solely on x, This yields more accurate estimate of xx and improves performance.
However, NsDiff also has several limitations. First, shown in Equation (8) the gliding-arriane, plays a crucial role, yet its estimator not effectively exploited. the reverse process, estimation is carried out by solving d univariate quadratic equations, rendering the sampling procedure unnec- essarily complicated. When solving these equations, failures may occur. To mitigate this issue, the sampling steps of reverse process should be set to relatively small value (e.g., 20) in order duce the probability of failure. Second, although NsDiff incorporates the diagonal sliding-variance Z it does not include the covariance, thereby ignoring correlations in multivariate time series. Third, in the reverse process of NsDiff, begins with Gaussian noise with variance 02 c This is
inconsistent with the terminal distribution whose variance is o%
B ALGORIHTSS OF JMCE, CW-DIFF AND CW-FLOW
The training procedure of JMCE is summarized in Algorithm T The training and sampling routines of CW-Diff are presented in Algorithms and respectively. Similarly, the corresponding training and sampling procedures for CW-Flow are provided Algorithms and and
C THEOREMS AND PROOFS
C.I THEOREMS
Theorem Define the Bregman divergence (Harandi et al. 2014, between two matri- M,, M2 of the same dimension as B(M,, M2)
(KL,NO, I] M2))
0.5 (Tr(M,2 M, I1a)- log Mi). Let Mxic Rddxdd be positive-defnite matrix and Mxic be a mnsitive-definie estimator of Mx Let Mxjc E [Mxics [xic }. A sufficient condi- tion for the inequality DKL (Pxic Q) < DKL PRu(Pxx (Pxic || N(Txic MMIe)) to hold is
|xxic 1 ||5xic-
Exto + |||0
|1
|
+ ( + d, M |x] - Mx
||=xcc + Mxiell M $2B(Ex Mxic)
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Algorithm Joint Mean-Covariance Estimator (JMCE)
Input: (Xn, C) in training set, hyperparameters Amin; Wiggen-
Output: A joint mean-covariance estimator JMCE(-).
1: Calculate sliding-window window covariances Exo.1,.... Exo.Ty of Xo
2: Initialize a non-autoregressive model J.CE(-)
3: while not converge do
4: Calculate Fx I.j..... LTTICC JMCE(C)
5: for = -, Tÿy do
6: Let E _x,,IC :.:0= T.j.C.-fcc
7: Perform eigen-decomposition of S Exx,tIc and obtain eigenvalues Exgine..s 1, 1,...,,
8: Perform singular value decomposition (SVD) of Ex., Exn,t/c and obtain singular values
Bi,e, ,,...,d
9:
end for
||o0 -xxc (L -i= Exo.t Exo.1c// LSVD =E'11 E $1,0
10: Calculate L2
11: RAm. -E Σ ReLUU(Ami - - (XEx).iic.)
Calculate LJMCE L2 + LSvD + Amiu / TyLF Wrigenn RA
Calculate LIMER and update the parameters of JMCE(.)
14: end while
return JMCE(-)
Algorithm Train Conditionally Whitened Diffusion model (CW-Diff)
Input: (Xo, C) in training set, diffusion schedule 87, E [0, 1]. a JMCE model JMCE(-).
Output: trained neural network s.W.
Calculate Txic: T.,.c....ITTAII = JMCE (C)
2: Calculate Σx, c T., IT,,ICTTT,
3: Calculate 5-0.5 Xa (1Tc) -..s ....,
4: Calculate XCW Exle (Xo- px(c)
5: Initialize neural network sew
6: while not converge do
7: Draw U10, 1]
8: Draw e ~ NN,, TaxdxT,)
9: Calculate a, = exp {- J0 8,ds/2) and = a?
Calculate LDIFF || CW (a,x1 + o,,,, C,T, E/O,
Calculate Vo and update the parameters of $9
12: end while
13: return sp
Theorem characterizes when replacing the terminal distribution N (āxjo, with Q leads to a reduction in the KLD between Pxic and the terminal distribution. This reduction occurs when
The estimator pxic closely approximates the true conditional mean μx|C-
Mxje is a reliable estimator of Mxics with the eigenvalues of M Mxjc bounded away from both zero and infinity.
The conditional covariance matrix Σxic' has eigenvalues bounded away from and infinity, deviates from Mxic. and is better approximated by well-estimated Exic than by M Mxic- This deviation formally measured by the Bregman divergence.
Setting Mxjo Mxjc Ia, and excluding Mxia in Theorem delineates the scenarios where our proposed replacement improves upon TMDM et 2024). Similarly, taking Mxic x
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Algorithm 3 Sampling from a trained CW-Diff model
Input: Historical observation test set, diffusion schedule Br, E [0, 1], JMCE model JMCE(.). trained neural network sy early stopping time Tmin- Output: Samples approximate Px/c.
IT, 2: Calculate Ex, = [I1cc|iic L,,,cTTy cc
1: Calculate Pxic. Ic JMCE(C)
3: Calculate 50.5 Cx =
(IT,,cITTTIc)) 10.9
4: Draw XCW ~ ~N(0, IdxdxTT
5: Solve SDE dxcw | 2 X.W. Brs0 t (XCW C.+) dr /B,aW, from 1 to
Tmin+ and get XCW
Tinnn
6: return Su.5 XCW + FxIc
Xo/C
Algorithm Train a Conditionally Whitened Flow Matching (CW-Flow)
Input: (Xo, C) in training set, JMCE model JMCE(-).
Output: trained neural network c.
1: Calculate exie, Lijc..... ...e....IIIII.. JMCE(C)
2: Calculate Σx, = [ Lr ITT
3: Calculate 50.5 "No [[_,cc (Tc))
(IT,,ICTT,
4: Initialize a neural network "
5: while not converge do
6: Draw U(0. 1]
7: Draw cw ~ (0, LaxdxT,)
8: Calculate cCW 0..5
C e CW + fx c
9: Calculate LFWw = || Xo - CW
(No + T( -oo), C,
Calculate T.ww and update the parameters of cW
11: end while
12: return CW
V
Algorithm 5 Sampling from a trained CW-Flow model
Input: Historical observation C in test set, JMCE model J.CE(-), trained neural network CW an early stopping time Tuun-
Output: Samples approximate Pxcc.
1: Calculate Pxie; . LT,ICC JMCE (C)
2: Calculate Σx, C [[ ....... Ic _
3: Calculate Su.s = [(21ccIT(c)
(LT, 10.5
Xx
4: Draw XCW NNO, LdxdxT;) Su.5 o X(W + Piic
5: Calculate XCW
Calc
6: Solve ODE cx - (XeW, C,r)dr from t = to Tiin, and get CW
Tmin
7: return CW Fmin
the matrix that contains only the main diagonal elements of Exic, and Mxic Mxic = c+ positive-definite diagonal estimator of o xx identifies cases where our method provides advantages NsDiff (Ye al. 2025). In practice, Exiс rarely coincides with la, xx particularly in
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time series data, where non-stationary dynamics (Li et al. 2024 Ye et al. 2025 and inter-variable dependencies (Yuan & Qiao. 2024) induce systematic departures from Id, or o x|c
C.2 THE PROOF OF THEOREM
In this section, we demonstrate how to establish the sufficient conditions for DKL (Px l| - DKL (Piie Qn). This sufficient condition fundamentally based on the following lemma.
Lemma Cardoso 2003) Let Pxic be a conditional distribution of X R given C, with conditional mean joycc: and conditional covariance Ex|c. For any Gaussian distribution Q = N (u, E), (Pxic Il Q) is given by:
DKL (Pxje II)) = DKL (Pxic |I N (uxic, Ex(o)) + DKL (N (uxic: Exie) I1)).
(10)
This is Pythagorean theorem for KLD. It tells us that the closest distribution to Pxic within the Gaussian family is Q. = (uxic Exie). Note that Pxic is not necessarily Gaussian. This also lays the foundation for our subsequent theoretical analysis. With (TO), we can rewrite
2 D Q)) DxL. (P Qo) as:
2 DKI 2) DrI, Qo)]
[Dka (Q= @) DQ 110e+11000 Qoi]
= -|5xic (mxic -ix 13 -11xxic112
(a)
+ log Exio| + Tr (Ex]c (ExicExic) Ir (Exjc).
(b)
As result, 2 (Pxic ②- - D (Pxic Qn) is decomposed into two parts: (a) and (b). In the following, we bound (a) and (b) separately. First, for (a), we have:
00 (Mxic xxxO 1: ||xxcll1
Ex 0.5 2
||exic- - -Axicll2 ||xxcil|2
max (xx(CH)) ) ||MxIc - |
min (xicatt | Moxic-xxii -xxc |
iI.....d
Then, derive upper bound of (b):
log
S x
+ ( (ExloXxie) - -Tr (Exic)
log
=
E Exic|
+ Tr (Ia,) Tr (Ta,) + Tr (Ex (ExleExic) Trr (Exic)
Σ 1 log log (1gggxx))))) - Tr (Exic) + Tr [Sxti (IXIC- ExIc)
$ Tr (Exic- -2x(ic + Ir Axa]
- (2x1c-
(2xic-
-Exic]
de
< Tr In (Exic- -Excc] + Σ X(CC(do-i-)) 3(0)
x-1
< 14 |Eicc- -Ex Excl,
(xte,
+ max i=l...,d
1=1
=o Exic Exic| +
(-min min (xiciat mmind
1=1
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where the first inequality comes from the bound + log S x for all x: O, the second inequality applies Von Neumann's 's trace inequality (Mirsky] 1975). and the third inequality uses the inequality which holds for any multiplicable matrices A and B.
Tr (AB) < || [|
Then, we derive:
[DralPxic (1 (Pxce Qw)
(1I)
(min ax cit) -1 [Il,...d
[ - 2xpal + Σ + ddz [xic - Ex1e| - ||x
1=1
which implies that as long as the right-hand side of 11 is non-positive (or equivalently, is true), we can have:
DRI. Q) s Dxa((xx c Qo).
C.3
THE PROOF OF THEOREM 2
By 10 we have
PRI(P (1)) DKL (i
2 [DKa 11 Q1 DKL (Q. ([ Mxie))
-Th - -a.)) + (uxic -ixic) T Exic (uxc- - -ixic)
log Ex Exlo - Tr ((Mxic- -Mxic ΣxIc)
-Tr -Ir(MiIOXiII 1d.) (uxic - -xic)) T Mx (pxic fxxic)
+ log M Ix Mxic + log Mx
s (uxic xxxic _
xxcc -Mxie (uxIc - -ix(c) + 2B(ExI ExIc)
+ +l|Mxic - Mxic
|Eiic + 1 -2B(ExIc, Mxic)
p
s ||xicc - - (11
||= - ix/0| + +|pxxcc| 1, 2 + + + 2x(xx
+ dd | Mx'cll Exic + Mxjalla 2B(Exio; Mxje);
where the first incquality follows from the matrix inequality log |M| s Tr(M Ia.). which holds when all cigenvalues of M are positive real numbers. This implies that long (9) is true, can have:
DKu(PxIe |I @) < DKL (Pxjo ( Mx(e)).
C.4 THE TERMINAL DISTRIBUTION OF 6
In this section, we prove the terminal distribution of 6 Mixic Ex,1c). First, let Y, expí.6 S B,ds/2] (X, - Pxic), then we can derive:
dY, = d(X, - Txx(c) elo 9.ds/2 + (X, - Txjc) deJo B,ds/2
[- / (ixccdd= V3, 50.5 o dWW.] eSo 9,ds/2
+ (X, (ix(c) _eo 3.d0/2 1
= co 9,ds/2 v/,
Xx o dW,,
5.0.5
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Integrate d',, from 0 to πI and we get:
h dY, =Y, - Yoo =
edo 0,ds/2 /Br
500.5
Na/C dWW,
Via the property of integral, we can derive:
Yn Yo ~o (0. /-'² eld cli 9.ds -Bdd ExxIc)
(0.1/-
equivalentl::
eSo B,ds/2 (X, - -xxic) ~N (Xo- - ixics (elō Aodo - 1). Ex,ic)
Finally, we can derive:
X, ~ N (Dxic e
-1-6" e- S0 B.da/2 (Xo (xx-(xxc), (1 - So Sadoy. Exo(e)
Recall exp(So $gds} becomes sufficiently large when + 1, then can derive that the terminal distribution X, is (Fx ()
D WHEN CAN INCORPORATING A PRIOR MODEL FAIL?
In this section, we discuss regimes in which Condition (3 may fail; in these regimes, incorporating prior model may degrade performance. We also describe how mitigate these risks.
First, if Pxic = O, then the right-hand side in 3 of (3) is dominated by estimation error. In this case, even small estimation errors cause the inequality in (3) to fail. In practice, however, for non-stationary time series, pX|C often exhibits sharp variations and thus deviates from zero, so this scenario unlikely to occur.
||xxcl| equals zero, while the left-hand side
Second, When mi...() Axic, is very small, the factor (min;e(1,...., Xxics) in (3 can
blow up, so modest deviations ||xi 7x cll2 and E Ex a IN may violate the dition. This motivates the explicit eigenvalue regularization in our JMCE loss in (4), which enforces strictly positive eigenvalues bounded away from zero
Finally, Condition 3 explicitly involves the estimation errors c ||x ExicllN. and ||E Excelr. p. If (fxic, Exic) are not good estimators of (μxie: Exic). then these terms on the left-hand side of (3) become large and may easily exceed || the right-hand side. To mitigate this, we deliberately design JMCE as a joint conditional mean and covariance estimator whose loss directly mirrors the left-hand side of (3). The experiments in SectionB.44sshe that JMCE achieves small estimation error, providing empirical evidence that our estimator yields sufficiently accurate (ūxic, Exic).
E EXTRA EXPERIMENTS
In this section, we first introduce the datasets and evaluation metrics, then report the performance of CW-Gen versus the Raw method the ETTh2, ILI, Weather, and Solar Energy datasets in Tables 3 In addition, we present the ProbMSE and ProbMAE of our CW-Gen against the Raw models in Table and We also conduct ablation studies show the effectiveness of our JMCE, as introduced in Section 3.2
E. DATASETS
We selected five widely used public real-world time series datasets for our experiments. ETT (Elec- tricity Transformer Temperature) dataset (Zhou et al.) 2021) contains hourly oil temperature and related external features (e.g.. load, ambient temperature) collected from electricity transformers between July 2016 and July 2018. We use two subsets, (1) ETThl and (2) ETTh2, which cover
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seven transformer-related factors. (3) ILI (Influenza-Lise Like Illness): Collects the weekly proportion of patients with ILI among all patients, which is reported weekly by the Centers for Disease Control and Prevention of the United States from 2002 to 2021. (4) Weather: A meteorological time series dataset collected from 21 weather stations in Germany, containing meteorological variables such as temperature, humidity, and wind speed recorded every 10 minutes. (5) Solar Energy: Records solar power generation data from 137 photovoltaic plants in Alabama, sampled every 10 minutes during 2006. The basic statistical information of these datasets are summarized in Table
E.2 METRICS
We employ six metrics to evaluate probabilistic time series forecasting. Among them, CRPS (Math-
eson & Winkler 1976) and QICE (Han al. 2022) are widely used. Let genn.[]] R k
1,..., K denote K generated samples. Denote X gen. and Xi as the (t, t)-th elements of X gen.[M] and Xn. respectively, for i = 1....,d d and t = I,...TTy Ty. The defination of CRPS between the K generated samples and Xg is given by:
Ty
CRPS( (en 1E No) d.
Xo)
1[Xb' s $
where F(.(2)= = # E 11x gen,[s] S 2} and }{{} is the indicator function.
To calculate QICE, we first construct equal quantile intervals from the generated samples (in our application, choose B = 10). In the ideal case, each interval should contain exactly 1/B of the entries of Xo. We then calculate the empirical frequency l of Xo 's entries falling into the b-th interval for = l,..., B. Finally, the QICE is calculated as:
QICE = B Σ
B
6m
Correlation score (Ni et al. 2022 measures the discrepancy between the correlations among the dimensions of the generated and the true time series. The covariance between the i-th and j-th features of Xo defined
Ty
Ty
cov,,j (Xo)
X
T
T
1=1
The Correlation score between Xo and X Xgen.[k] |k] is defined as:
1 d d
Σ
covi,j (Xo)
COV; (X Xgen, (k))
Correlation score(X (Xenn(k)) Xn)
cov (Xn)oov (Xn)
cov,,; (X gon, jeov,oovo (X
1.j
The Probabilistic Correlation Score (ProbCorr) is defined as:
K
ProbCorr((Xxco ] 1= Xo) =
Σ
Correlation score(X, (Xgen Xo).
K
k=1
ProbCorr measures the discrepancy between the correlation structure of each generated sample Xgen,(k) and the ground-truth Xo-
Nevertheless, it is important to recognize that CRPS, QICE, and ProbCorr do not effectively capture temporal dependencies between the generated and true sequences. To address this, a TS2Vec model (Yue et al. 2022) is trained on the real sequence [C, Xo] and subsequently used to extract latent representations for [C, Xu) and [C, Xgengen,[s)]] h = 1.... K. The Fréchet Inception Distance (FID) computed between these representations referred to the conditional FID. Since TS2Vec em- ploys a dedicated network architecture to jointly capture temporal patterns and feature correlations, conditional FID provides more comprehensive assessment of generative quality.
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In addition to probabilistic metries, we also employ ProbMSE and ProbMAE to evaluate forecasting performance. ProbMSE and ProbMAE are used as point forecast metrics, and their definition is given by:
Ty
ProbMSE( (X [e).
xi
T
gen,[k))
ProbMAE X gon, 1 Xo)
X
X
d T
gen,
Traditional MSE and MAE measure the discrepancy between the mean of the generated samples and the true time series. In contrast, ProbMSE and ProbMAE differ from these traditional metrics by taking into account the MSE and MAE between each individual generated sample and the true time series. Consequently, ProbMSE and ProbMAE provide a more stringent evaluation than standard MSE and MAE.
E.3 CW-GEN ON MORE REAL DATASETS
Tables 3 5 and 6 present the results of different methods on the ETTh2, ILI, Weather, and Solar Energy datasets across four metrics (CRPS, QICE, ProbCorr, and Conditional FID). In time series forecasting tasks, ProbMSE and ProbMAE reflect the accuracy of point estimates. We report the evaluation results on these two metrics for the five real-world datasets in Tables and respectively.
From Table to S we can observe that CW-Gen achieves advantages on the majority of probabilistic metrics,
digeesinnsi datasets such as Solar Energy. For point forecasting metrics, CW- Gen outperforms the baselines on all datasets except Solar Energy.
Table 3: Metrics for models trained on original ETTh2 (Raw) and conditionally whitened ETTh2 (CW). Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates of every metric of Raw and CW-Gen models also provided.
Model
CRPS (L)
QICE (L)
ProbCorr (L)
Conditional FID (L)
(ETTh2)
Raw
CW
Raw
CW
Raw
CW
Raw
CW
TimeDiff
2.543
0.395
12.769
6.584
0.753
0.327
211.67
4.495
2023)
(0.910)
(0.031)
(1,726)
(1.246)
(0.252)
(0.020)
(55.976)
(0.699)
SSSD
0.754
0.458
14.698
6.637
0.525
0.417
187.29
14.780
2023
(0.260)
(0.111)
(0.955)
(3.059)
(0.040)
(0.039)
(147.33)
(7.330)
Diffusion
1.107
0.381
8.605
4.147
0.691
0.438
99.509
15.383
-TS (2024)
(0.077)
(0.024)
(0.792)
(1.677)
(0.022)
(0.061)
(64.135)
(16.112)
TMDM
0.421
0.377
4.500
3.945
0.378
0.313
9.528
4.107
2024
(0.043)
(0.000)
(0.689)
(1.475)
(0.027)
(0.001)
(1.779)
(0.249)
NsDiff
0.370
0.369
2.334
2.579
0.323
0.351
19.957
14.842
2025
(0.027)
(0.014)
(0.040)
(0.345)
(0.026)
(0.018)
(5.029)
(2.783)
FlowTS
1.534
0.824
147
11.744
0.650
0.498
80.540
10,640
2025
(0.252)
(0.138)
(1.356)
(1.094)
(0.044)
(0.050)
(69.867)
(12.883)
Win rate
0.0%
100.0%
16.7%
83.3%
16.7%
83.3%
0.0%
100.0%
E.4 ABLATION STUDY FOR JMCE
In this subsection, we examine the impact of the two hyperparameters WBigen and Amin in (4p esti- mation accuracy of JMCE. Beside, we also investigate the influence different backbones on JMCE. While the main text employs the Non-stationary Transformer (Liu et al.) (022), in this section adopt FED Former (Zhou et al. 2022) and Informer (Zhou et al. 2021) as the backbones of JMCE and compare their performance.
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Table 4: Metries for models trained on original ILI (Raw) and conditionally whitened ILI (CW).
Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates of every metric of Raw and CW- Gen models are also provided.
Model
CRPS (1)
QICE (L)
ProbCorr (L)
Conditional FID (L)
(ILI)
Raw
CW
Raw
CW
Raw
CW
Raw
CW
TimeDiff
1.148
1.046
15.015
13.597
0.455
0.399
10.957
6.845
(023))
(0.134)
(0.081)
(0.430)
(1.550)
(0.016)
(0.048)
(3.483)
(0.813)
SSSD
1.038
0.758
15.063
9.115
0.374
0.365
6.416
5.964
(2023)
(0.126)
(0.110)
(1.802)
(2.030)
(0.071)
(0.038)
(0.335)
(1.895)
Diffusion
1.222
0.769
6,588
8.883
0.381
0.373
20.513
5,969
-TS (2024)
(0.271)
(0.168)
(2.479)
(1.840)
(0.038)
(0.058)
(27.582)
(1.215)
TMDM
0.796
0.722
6.706
8.029
0.365
0.359
22.693
12.234
2024)
(0.045)
(0.025)
(0.821)
(2.734)
(0.020)
(0.000)
(12,420)
(18.767)
NsDiff
0.738
0.645
5,930
6.173
0.352
0.307
73.379
14,852
2025)
(0.047)
(0.059)
(0.867)
(0.970)
(0.011)
(0.058)
(23.257)
(3.843)
FlowTS
0.997
0.851
9.771
10.645
0.413
0.410
7.689
6,202
(2025)
(0.055)
(0.068)
(0.728)
(0.778)
(0,010)
(0.021)
(1.098)
(0.536)
Win rate
0.0%
100.0%
66.7%
33.3%
0.0%
100.0%
0.0%
100.0%
Table 5: Metrics for models trained original Weather (Raw) and conditionally whitened Weather (CW), Each experiment repeated by 10 times, and standard deviations are provided brackets. The better results between Raw and CW are underlined. The win rates of every metric of Raw and CW-Gen models are also provided.
Model (Weather)
CRPS (L)
QICE (L)
ProbCorr (L)
Conditional FID (L)
Raw
CW
Raw
CW
Raw
CW
Raw
CW
TimeDiff
0.531 (0.032)
0.258
8.530
6.772
0.362
0.255
9.673
6,892
(003))
(0.014)
(0.693)
(1.869)
(0.010)
(0,006)
(3.095)
(1.183)
SSSD
0.499
0.530
7.428
4.121
0.438
0,411
914.81
330.31
(2023)
(0.145)
(0.186)
(0.790)
(3.457)
(0.012)
(0.040)
(260.690)
(315.18)
Diffusion
0.495
0.319
3.957
3.047
0.503
0.414
278.60
90.739
-TS (2024)
(0.114)
(0.033)
(7.789)
(1.049)
(0.033)
(0.047)
(566.85)
(59.186)
TMDM
0.231
0.254
3.468
3.127
0.264
0.247
6.978
5.941
2024)
(0.003)
(0.016)
(0.412)
(0.733)
(0.008)
(0.010)
(0.836)
(0.860)
NsDiff
0.270
0.262
3.746
3.536
0.274
0.266
18,034
9,870
2025)
(0.003)
(0.009)
(0.201)
(0.408)
(0.007)
(0.007)
(0.887)
(3.936)
FlowTS
0.348
0.244
6,901
6.598
0.334
0.262
8.447
6.948
2025))
(0.043)
(0.017)
(1.616)
(0.371)
(0.035)
(0.011)
(1,540)
(2.887)
Win rate
33.3%
66.7%
0.0%
100.0%
0.0%
100.0%
0.0%
100.0%
We adopt L2, LF, LSVD in 4 as the metrics. In addition, also compute the left-hand side (LHS)
of 3 as a metric, whose formulation is given by:
LHS = (min A A (Lz + LSvD) + TT.y.
(12)
Tables and 10 report JMCE results under varying Wrigen and Amins respectively, and Table 21 presents CW- Gen results with separately trained models and different backbones.
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Table 6: Metries for models trained on original Solar Energy (Raw) and conditionally whitened Solar Energy (CW). Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates of every metric of Raw and CW-Gen models are also provided.
Model (Solar)
CRPS (L)
QICE (L)
ProbCorr (L)
Conditional FID (1)
Raw
CW
Raw
CW
Raw
CW
Raw
CW
TifeDiff
0.746 (0.017)
0.299
15.361
11.998
0.198
0.212
5.606
4.397
2023)
(0.016)
(0.617)
(1.383)
(0.004)
(0.007)
(0.357)
(0.232)
SSSD
0.350
0.555
13.435
9.111
0.330
0.307
14.915
14.165
(2023)
(0.042)
(0.135)
(1.966)
(0.675)
(0.020)
(0.081)
(0.622)
(4.571)
Diffusion
0.349
0.289
2.857
2.843
0.229
0.221
5.796
5,007
-TS (2024)
(0.030)
(0.026)
(1.326)
(1.186)
(0.008)
(0.008)
(0.665)
(0.902)
TMDM
0.376
0.371
10.033
7.162
0.509
0.201
248.80
8.279
2024)
(0.004)
(0.018)
(0.076)
(0.214)
(0.008)
(0.012)
(16.384)
(2.528)
NsDiff
0.304
0.359
6,861
4.251
0.366
0.239
106.83
7.417
2025)
(0.008)
(0.007)
(0.318)
(0.345)
(0.011)
(0.015)
(8.575)
(1.051)
FlowTS
0,276
0.226
6.791
4.207
0.284
0.210
28.464
3.225
(2025)
(0.029)
(0.018)
(0.687)
(1.162)
(0.025)
(0.011)
(5.609)
(0.157)
Win rate
33.3%
66.7%
0.0%
100.0%
16.7%
83.3%
0.0%
100.0%
Table 7: ProbMSE for Raw and CW-Gen models. Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates for all datasets are also provided.
Model
Variant
ETThl
ETTh2
ILI
Weather
Solar
TifeDiff
Raw
1.366(0.080)
0.793(0.983)
3.803(2.06) (2.62)
0.803
0.800().020)
2023
CW
(.756(0.135)
0.496(0.064)
2.913(0.303)
0.267(0.014)
0.264(0.002))
SSSD
Raw
1.493(0.390)
2.1((00.82
2.953(0.419)
2.785(3.243)
0.349(0.079)
2023
CW
0. 0.908(0.219)
0.643(0.2622)
2.169(0.467)
2.158(2.761)
L.053(0.508)
Diffusion
Raw
.177(0.094)
2.053(L.07 .078)
2.224(0.497)
1.287(0.322)
0.391(0.029)
-TS
CW
0.717(0.094)
(.503(0.071)
2.788(0.658)
0.34500.855)
0.326(0.045)
TMDM
Raw
0.767(0.070)
0.615(0. 118)
2.417(0.189)
0.249(0.007)
0.243(0.014)
2024]
CW
(.681(0.010)
0.488(0.001)
(.984(0.113)
0.284(0.024)
0.418(0.065)
NsDiff
Raw
0.637(0.075)
0.649(0.040)
2.424((.163)
0.283(0.008)
0.277(0.021)
2025
CW
0.729(0. 132)
0.488(0.041)
1.759(0.224)
0.292(0.012)
0.507(0.030)
FlowTS
Raw
1.006(0.153)
2.958(0.774)
2.960(0.250)
0.455
0.262
2025
CW
0.698(0.059)
L.522(0.429)
2.369(0.254))
0.272(0.029)
0.240(0.024)
Win Rate
83.33%
83.33%
100.0%
66.67%
50.00%
Table shows that as WFigen increases, the smallest eigenvalue moves further away from zero, which aligns with the intended purpose of this parameter. Surprisingly, the estimation of both the condi- tional mean and the sliding-window window covariance also becomes more accurate with larger W'ggen-
Table 10 shows that as Amin increases, the smallest eigenvalue moves further away from zero A larger Amin leads to poorer estimation of the sliding-window covariance, because the features of real-world time series typically highly correlated and thus the sliding-window covariances have very small minimum eigenvalues. Penalizing the eigenvalues with larger Amia alters the structure of the estimation.
Table showss that separately training the estimator of the conditional mean and that of the sliding- window covariance does not effectively control the smallest eigenvalue, although this training strat-
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Table 8: ProbMAE for Raw and CW-Gen models. Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates for all datasets are also provided.
Model
Variant
ETTh1
ETTh2
ILI
Weather
Solar
TimeDiff
Raw
0.899(0.040)
0.480(0.055)
1.025(0,436)
0.670(0.034)
0.800(0.021)
2023
CW
0.489(0.033) (..81(0.025)
1.121(0.076)
0.307(0.009)
0.331(0.019)
SSSD
Raw
0.959(0. 150)
1.512(0. 140)
(0.00)))090)
1.190(0.533)
0.294(0.038)
22023
CW
0.704(0.069)
0.581 100)
(.941(0.157)
0.755
0.685(0. 164)
Diffusion
Raw
0.801().037)
1.891(0.537)
1.018(0.135)
0.884(0. 125)
0.451(0.028)
-TS (2024)
CW
0.594(0.031) 0.508(0.045)
1.178(0.222)
0.381(0.066)
0.390(0.039)
TMDM
Raw
0.627(0.039)
0,551(0.054)
0.990(0.044)
0.294(0.003)
0.303(0.008 (060
2024
CW
0.503().047) 0,484(0.000)
0.899(0.011)
0.325(0.020))
0.430(0.021)
NsDiff
Raw
0.557(0.032)
0.544(0.016)
1.005(0.063)
0.325(0.005)
0.345(0.013)
2025
CW
0.553(0.022)
0,481(0.019)
0.812(0.068)
0.330(0.009)
0.458
FlowTS
Raw
0.742(0.079)
1.01900.44)
1.039(0.088 0844
0,447(0.056)
(.32400.33)
2025
CW
0.598(0.021)
0.946(0. 142)
0.961(0.066)
0.303(0.022)
0.274(0.023)
Win Rate
100.00%
83.33%
100.0%
66.67%
50.00%
egy offers slight advantage in estimating the conditional mean. Different backbones also lead to different outcomes. FED-Former performs well in estimating the conditional mean but slightly less effective in estimating the sliding-window covariance. In contrast, Informer achieves strong performance in covariance estimation and eigenvalue control, yet performs the worst in estimating the conditional mean.
In addition, we provide a comparison between the learning targets and the outputs of JMCE in Fig- 4 As shown in the figure, JMCE able accurately predict both the future time series and most components of the sliding-windov covariance on training and test sets. For some diagonal compo- nents of the sliding-window window covariance (such as Cov Dim 8, 19, 26, and 28), JMCE intelligently enlarges these values, which helps prevent the minimum eigenvalue from becoming too small.
Table 9: Metrics for JMCE trained on ETTh1, with different Whigen- The Amin is set to 0.1. Each experiment is repeated 10 times and standard deviations provided in brackets. The defination of LHS can be found in (22
(mine, A5X
Model
L2 (L)
CF (L)
LSVD (J)
(4)
LHS (L)
JMCE
0.702
0.198
0.493
2.2 107
2.4 - 107
(WEigen 0)
(0.006)
(0.000)
(0.000)
(4.4 1015)
(5.5 1025)
JMCE
0.672
0.186
0.526
14.472
24.197
(WEigen 10)
(0.002)
(0.000)
(0.000)
(4.540)
(6.258)
JMCE
0.683
0.189
0.534
12.996
22.819
(WEigen
20)
(0.005)
(0.000)
(0.000)
(1.438)
(4.422)
JMCE
0.693
0.184
0.529
12.175
21.651
(W'nggen 40)
(0.005)
(0.000)
(0.000)
(6.609)
(13.283)
JMCE
0.726
0.180
0.528
11.487
21.037
(WEigon
50)
(0.001)
(0.000)
(0.000)
(3,286)
(5.085)
E.5 INFLUENCE OF DIFFERENT JMCES ON CW-GEN
In this subsection. we investigate the impact of different JMCE models on CW-Gen. We first train the JMCE with different hyperparameters, JMCE with different backbones, separately trained
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ETThl
TimeDiff
SSSD
Diffusion-1
TMDM
NsDiff
Sif
CW-SSSD
CW Diffusion TS
CW-TMDM
CW-NsDill
CW-FlwwI
ETTh2
Dife
SSSD
Diffusion-TS
TMDM
NsDift
FloWTS
CW TimeDiff
CW-SSS) SSSD
CW Diffusion-1 TS
CW TMDM
CW-N-DiI
CW FlowTS
ILI
TimeDiff
SSSD
Diffusion-TS
TMDM
NsDiff
FlowTS
Diff
CW-SSSD
CW-Diffusion-TS
CW-TMDM
CW-NsDiff
CW-FlowTS
Weather
TimeDiff
SSSD
Diffusion TS
TMDM
N:Diff
FlowTS
CW-TimeDiff
CW-SSSD
CW-Diffusion-TS -T
CW-TMDM
CW-N-Diff
CW-FlowTS
Figure 3: Comparison of all models on ETTh1, ETTh2, ILI and Weather.
mean and covariance models. Then, these models are severed as the prior for the CW-Gen models. The CW-Gen models are evaluated by the same metries as Section
Table 12 and Table 13 indicate that the default parameters in our paper (WEigen 50, Amin 0.1) achieve slight advantages over other parameter combinations. Table 21 shows that CW-Gen models using different JMCEs exhibit slight variations, but all CW-Gen models with JMCE priors outperform those with separately trained prior models in most cases. Among the three backbones, the Non-stationary Transformer achieves the best performance on 13 metrics, while FED-Former achieves and Informer achieves 1. Therefore, we adopt the Non-stationary Transformer as the backbone of JMCE.
E.6 CONTRIBUTIONS OF INDIVIDUAL COMPONENTS OF THE PRIOR MODEL
In this section, we investigate how different components JMCE contribute to the improvement. Specifically, only the conditional mean (Mean), only the diagonal elements of the condi-
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MAN
ittfi
Figure 4:
Comparison between the learning targets and the predictions JMCE (top: training set, bottom: test set).
tional covariance (Var), only the full conditional covariance (Cov), well the combination of the conditional mean and the conditional variance (Mean & Var) as prior information, and then evaluate the performance of CW-Gen. Our default CW-Gen adopts the conditional mean together with the full conditional covariance (Mean & Cov) as the prior.
According to Table 14
we can generally conclude that using only the conditional mean for center- ing yields slightly inferior performance, compared to whitening using the conditional mean together with conditional variance or covariance. This indicates that learning the conditional variance or covariance of the time series provides beneficial prior information for the generative model. More- CW-Gen when using the conditional mean together with
over, if we compare the performance the conditional variance versus using the conditional mean together with the conditional covariance, we observe that in most cases CW-Gen performs better with the full conditional covariance. This suggests that incorporating the full conditional covariance, rather than only the variance, provides stronger and more informative prior for the generative model. In addition, using only the conditional variance only the full conditional covariance as the prior degrades the performance of CW-Gen.
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Table 10: Metries for JMCE trained on ETThI, with different Amin- The Wigen 50. Each experiment is repeated 10 times and standard deviations are provided in brackets. The defination of LHS can be found in
Model
L2 (4)
Cr (4)
LSvD (4)
mine, A5 Exp.1
LHS (L)
JMCE
0.665
0.193
0.484
2.3 -10
2.7 10a
(Amin 10-3)
(0.002)
(0.000)
(0.000)
(1.2 (1.2.10)
(1.6 10%)
JMCE
0.662
0.195
0.493
157.69
190.60
(Amin 10
(0.007)
(0.000)
(0.000)
(5.4 100)
(8.0-10%)
JMCE
0.680
0.195
0.509
23.573
35.213
(Amin = 0.05)
(0.004)
(0.000)
(0.000)
(3.423)
(7.417)
JMCE
0.726
0.180
0.528
11,487
21 21.037
(Amin 0.1)
(0.001)
(0.000)
(0.000)
(3.286)
(5.085)
Table 1: Metrics for JMCE separately or jointly trained on ETTh1, with different backbones. The Amin set and Wrigen set to 50. Each experiment is repeated 10 times and standard deviations provided in brackets. The defination of LHS can be found in 12). NS, FED, and IN indicate that the backbone of JMCE is the Non-stationary Transformer, TED-Former, and Informer, respectively.
Model
L2 (L)
Cp (4)
LSVD (L)
min, Z
LHS (L)
Separate
0.721 (0.100)
0,421
0.797
1196
1824
(NS)
(0.164)
(0.078)
(764)
(1184)
JMCE
0.726
0.180
0.528
11.487
21.037
(NS)
(0.001)
(0.000)
(0.000)
(3.286)
(5.085)
JMCE
0.548
0,387
0.741
43,381
70.206
(FED)
(0.026)
(0.047)
(0.030)
(12.833)
(16.962)
JMCE 1.224
0.260
0.609
10.481
28.821
(IN)
(0.039)
(0.015)
(0.016)
(1.605)
(2.956)
However, the latter achieves lower ProbCorr than the former, indicating that leveraging the full conditional covariance makes CW-Gen better capture the inter-variable dependencies.
E.7 ABLATION STUDY OF THE LENGTH SLIDING WINDOW
In this section, we compare the effect of the length of the sliding window. In our main experiments, we set the length of sliding window as 95, following NsDiff (Ye et al.) 2025). In ETThl dataset, we compare the performance of CW-Gen under four additional window lengths, namely 75, 85, 105, and 115. In ILI dataset, we compare two additional window lengths, namely 11 and 19.
From Table 15 we observe that on ETTh1, the sliding window length does not introduce substantial changes to the performance CW-Gen. In contrast, Table 16 shows that the sliding window has a somewhat larger impact the ILI dataset. This is likely because the dataset is relatively short, and changes in the sliding window length may alter the underlying dependence relationships.
E.8 CW-GEN COMPARED WITH OTHER UNIVARIATE PRIOR METHODS
We discuss the similarities and differences between CW-Gen and other univariate generative models that incorporate prior information, like DSPD (Biloš et al. 2023) and TsFlow (Kollovieh et al, 2025).
DSPD leverages kernel functions such as exp(->/lt-ta) tyl) and (xtp(-y(t-te) 3), > 0. to help the diffusion model better capture the temporal correlations restricted to the prediction window. How-
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Table 12: Metries for CW-Gen models with different Wigen of JMCE. Each experiment repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates of every metric for different Whigen are also provided.
Model
CRPS (L)
QICE ())
ProbCorr (1)
Conditional FID (L)
Wigen
40
50
40
50
40
50
40
50
TifeDiff
0.495
0,505
8.069
8.821
0.235
0.243
6.835
6.788
2023
(0.038)
(0.040)
(2.310)
(1.916)
(0.035)
(0.027)
(7.952)
(5.425)
SSSD
0.510
0.524
4.935
4.838
0.239
0.238
7.438
9.265
(2023)
(0.099)
(0.085)
(2.544)
(1.921)
(0.026)
(0.024)
(2.538)
(5.003)
Diffusion
0.447
0.445
2.333
2.963
0.276
0.266
11.913
7.686
-TS (2024)
(0.014)
(0.024)
(0.831)
(0.887)
(0.027)
(0.012)
(9.911)
(2.751)
TMDM
0.443
0.440
4.131
4.555
0.209
0.213
3.554
3.831
2024)
(0.000)
(0.001)
(1.128)
(0.855)
(0.000)
(0.001)
(0.283)
(0.431)
NsDiff
0.422
0,431
1.264
1.249
0.200
0.206
8.160
8,820
2025
(0.020)
(0.029)
(0.252)
(0.228)
(0.014)
(0.010)
(L.185)
(1.541)
FlowTS
0.491
0.488
9.014
8.817
0.261
0.254
5,030
4.865
(0255)
(0.033)
(0.020)
(0.313)
(0.460)
(0.020)
(0.021)
(0.871)
(0.563)
Win rate
50.0%
50.0%
33.3%
66.7%
50.0%
50.0%
50.0%
50.0%
Table 13: Metries for CW-Gen models with different Amin of JMCE. Each experiment repeated by 10 times, and standard deviations are provided in brackets. The better results between Raw and CW are underlined. The win rates of every metric for different Amin are also provided.
Model
CRPS (L)
QICE (L)
ProbCorr (4)
Conditional FID (4)
Amin
0.05
0.1
0.05
0.1
0.05
0.1
0.05
0.1
TifeDiff
0.508
0.505
9.101
8.821
0.230
0.243
5.527
6.788
2023)
(0.035)
(0.040)
(1.753)
(1.916)
(0.027)
(0.027)
(3.326)
(5.425)
SSSD
0,530
0.524
5.166
4.838
0.246
0.238
9.637
9.265
(2023)
(0.104)
(0.085)
(2.257)
(1.921)
(0.038)
(0.024)
(4.358)
(5.003)
Diffusion
0.453
0.445
2.760
2.963
0.271
0.266
10.553
7.686
-TS (2024)
(0.024)
(0.024)
(1.093)
(0.887)
(0.031)
(0.012)
(5,914)
(2.751)
TMDM
0.446
0.440
4.260
4,555
0.216
0.213
3.702
3.831
2024)
(0.001)
(0.001)
(0.785)
(0.855)
(0.001)
(0.001)
(0.475)
(0.431)
NsDiff
0.416
0.431
1.369
1.249
0.199
0.206
8.477
8.820
2025)
(0.030)
(0.029)
(0.256)
(0.228)
(0.021)
(0.010)
(1.934)
(1,541)
FloWTS
0.494
0.488
8.969
8,817
0.257
0,254
5.131
4.865
2025)
(0.038)
(0.020)
(0.557)
(0.460)
(0.026)
(0.021)
(0.717)
(0.563)
Win rate
16.7%
83.3%
33.3%
66.7%
33.3%
66.7%
50.0%
50.0%
ever, such prior information does not incorporate the historical time series and conditional mean. Therefore, DSPD does not provide stronger guidance for forecasting then JMCF. Our JMCE can explicitly capture the correlations between variables by directly learning the sliding-window covari- ance on the prediction window. Moreover, the architecture of the Non-stationary Transformer en- ables JMCE to capture temporal correlations within the prediction window via masked self-attention. It also captures correlations between the observed series and the prediction window through cross- attention (Liu et al. 2022).
TsFlow employs Gaussian processes (GPs) to predict the conditional mean and variance within the prediction window. However, GPs rely heavily on the choice of kernel functions, and modeling non-stationary processes typically requires carefully designed kernels kernels with varying length
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Table 14: Metries for models trained on ETThl conditionally whitened by different priors. Each experiment is repeated by 10 times, and standard deviations are provided in brackets. The best results are underlined and the second-best results are dashedd-udderlneed
Model
Prior
CRPS
QICE
ProbCorr
Conditional FID
TifeDiff 22033
Mean
0.503(0.037)
8.001 (49
0.245(0.034)
8.173(9.390)
Var
0.581(0.045)
0.235(0.014) 9.153(1.332)
7.385(2.153)
Cov
0.624(0.053)
9.133(1.557)
0.258(0.049)
25.858(50,988)
Mean & Var
0.512(0.032)
9.883(2.368
0.246(0.023)
6.749(7.313)
Mean & Cov
0.505(0.040)
8.821(1.916)
0.243(0.027)
6.788(5.415)
SSSD (2023
Mean
0.554(0.116)
7.175(2.386)
0.242(0.028)
6,431(2.537)
Var
0.566(0.080)
5.738(1.911)
0.349(0.043)
18.689 1199911448)
Cov
0.587(0.067)
5.190(1.410)
0.340(0.037)
18.903(13.821)
Mean & Var
0.530(0.087)
5.203(1.840)
0.237(0.015)
7.235(2.265
Mean & Cov
0.524(0.085)
4.838(1.921)
0.238(0.024)
9.265(5.003)
Diffusion
Mean
0.468(0.035)
2.544(0.897)
0.301(0.027)
8.535(2.208)
-TS (2024)
Var
0.536(0.064)
4.616(0.990)
0.474(0.045)
68.466(43. 107)
Cov
0.549(0.032)
5.360(1.176)
0.502(0.046)
95.769(50.600)
Mean & Var
0.465(0.027)
2.539(..182)
0.288(0.026)
8.558(3.331)
Mean & Cov
0.445(0.024)
2.963(0.887)
0.266(0.012)
7.686(2.751)
TMDM 2024)
Mean
0.446(0.000)
3.760(1 201)
0.215(0.000)
4.687(1 .447)
Var
0.644(0.002)
(.372(1.263)
(.291(0.000)
14.107(9.783)
Cov
0.676(0.001)
6.377(1.237)
0.303(0.000)
(7945(199)543)
Mean & Var
0.440(0.000)
4.952(0.911)
0.213(0.000)
3,840(0.450
Mean & Cov
0.440(0.001)
4.55(0..855)
0.213(0.001)
(0.31)
NsDiff 22025)
Mean
0.455(0.023)
4.647(0.500)
0.238(0.006)
71.683(9.262)
Var
0.623(0.011)
4.132(0.450)
0.209(00011)
23.738(3.654)
Cov
0.409(0.022)
1.857(0.290)
0.19((0.013)
11.440(.2000
Mean & Var
0.408(0.024)
1.447(0.259)
0.209(0.012)
21.288(3.455 4566
Mean & Cov
0.431(0.029)
1.249(0.228)
0.206(0.010)
4880(1.541)
FlowTS 20055
Mean
8.778(0.798) 0.477(0.024)
0.231(0.010)
4.850
Var
(0.02)
8.328(0.701)
0.316(0.010)
10.459(2.206)
Cov
0.643(0.020)
8.064(0.634)
0.315(0.005)
10.344(2.64)
Mean & Var
0.489(0.022)
9.240(0.477)
0.260(0.027)
5.090(0.250)
Mean & Cov
0.488(0.743)
8.817(0.460)
0.254(0.02))
4.865(0.563)
scales, making the approach less straightforward in practice. Moreover, computing the GP and variance requires inverting a matrix whose size equals the length of the historical observations. In our setting, the GP requires inverting matrix in Th xTr, which incurs computational complexity of OTT)) which is higher than our JMCE when Tí > d.
In addition, we apply DSPD and TsFlow each individual dimension of ETTh and generate sam- ples accordingly. We then aggregate the generated samples and evaluate them using the four metrics introduced in Appendix E.2 In Table I7] we report the performance of DSPD and TsFlow. Com- pared with CW-TimeDiff and CW-SSSD from the same year (2023), DSPD exhibits worse CRPS and high QICE, ProbCorr of DSPD also lower than both models, while Conditional FID lies at an intermediate level. Besides, compared with CW-NsDiff and CW-FlowTS proposed in 2025, TsFlow shows worse CRPS, QICE, and ProbCorr, while its Conditional FID lies between the two. Their performance on the first dimension of ETThl is further illustrated in Figure As observed, DSPD lacks the prior information provided by the conditional mean and therefore fails to effec- tively capture highly nonlinear patterns. On the other hand, TsFlow produces results that less stable near the end of the prediction window, indicating that its effectiveness is limited long-term forecasting settings.
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Table 15: Metries for models trained on ETThI, with different length sliding window. Each experiment is repeated by 10 times, and standard deviations are provided in brackets.
Model (ETTh1)
Window
CRPS
QICE
ProbCorr
Conditional FID
Dife
75
0,477(0.030)
8.039(2.242)
0.232(0.025)
4.858(1
(2023)
0.511(0.036)
9.249(1.610)
0.249(0.044)
6.236(3.83)
95
0.505(0.040)
8.821(1.916)
0.243(0.027)
6.788(5,4155
105
0.518(0.044)
8.548(1.539)
0.249(0.021)
7.780(5.61)
115
0.502(0.043)
7.621(1.925)
0.251(0.037)
4.750(1.318)
SSSD
75
0.508(0.072)
4.525(2.263)
0.235(0.024)
7.531(2.885)
(2023
85
0.520(0.070)
4.484(2.086)
0.235(0.021)
8,236(2.754)
95
0.524(0.085)
4.838(1.921)
0.238(0.024)
9.265(5.003)
105
0.540(0.140)
4.383(2.391)
0.248(0.018)
7.858(2.501)
115
0.519(0.069)
4.392(2.062)
0.249(0.021)
9.189(2.733)
Diffusion
0.425(0.012)
2.439(0.767)
0.248(0.023)
10.527(11.100)
-TS 2024
85
0.431(0.020)
3.084(1.712)
0.247(0.022)
11.319(1 1,244)
95
0.445(0.024)
2.963(0.887)
0.266(0.012)
7.686(2.751)
105
0.452(0.030)
2.876(1.388)
0.256(0.022)
7.639(3.425
115
0,450(0.024)
2.473(1.110)
0.268(0.029)
9.613(5.99)
TMDM
75
0.429(0.000)
4.307(0.593)
0.213(0.000)
3.789(0.031)
(024)
0.435(0.000)
4.398(0. 760)
0.212(0.000)
3.622(0.059)
95
0.440(0.001)
4.555(0.855)
0.213(0.001)
3.831(0.431)
105
0.464(0.000))
4.655(1.288)
0.231(0.000)
4.036(0.168)
115
0.442(0.000)
3.981(0.475)
0.225(0.001)
3.631(0.280)
NsDiff 2005
75
0.429(0.023)
1.210(0.266)
0.203(0.013)
9.846(3.176)
85
0.423(0.021)
1.193(0.203)
0.203(0.009)
9.025(1.011)
95
0.431(0.029)
1.249(0.228)
0.206(0.010)
8.820(1.541)
105
0.422(0.020)
1.281(0.212)
0.207(0.01))
9.707(1.354)
115
0.432(0.024)
L.376(0.228)
0.213(0.022)
10.030(2.669)
FlowTS (2025)
0.484(0.023)
8.965(0.388)
0.253(0.01L)
4.886(0.426)
85
0.476(0.015)
8. 8.865(0.340)
0.251(0.015)
5.016(0.637)
95
0.488(0.020)
8.817(0.460)
0.254(0.021)
4.865(0.563)
105
0.483(0.024)
8.894(0.704)
0.256(0.010)
4.920(0.364)
115
0.475(0.020)
8.924(0.583)
0.253(0.015)
4.813(0.456)
E.9 ACCELERATING CW-GEN
Training a CW-Gen model consists of three steps: (1) training the JMCE model, (2) conditionally whitening the time series using the trained JMCE model, and (3) training the generative model by the conditionally whitened time series.
In step (I), the training algorithm of JMCE involves SVD and cigen-decomposition, both of which have computational complexity of O(d)). Although these operations can be efficiently parallelized on GPUs, they still pose challenges when implementing the model on high-dimensional time-series datasets. Under high-dimensional cases, and the penalty term RA.... in (4) In this case, to ensure that the minimum eigenvalue remains bounded away from zero, we can add Amin Ta either to JMCE's output E.jc or to Ex,,(C- Another possible approach is to only learn the diagonal part the diagonal covariance matrix cannot approximate a general covariance matrix well in terms of the nu- clear norm. Besides, diagonal covariance parameterizations lose the ability to control the minimum eigenvalue of the conditional covariance matrix, and therefore the theoretical foundations of JMCE no longer apply. Empirical evidence in Table[4[alss a4jass indicates only using conditional variance leads to inferior performance, especially for ProbCom..
possible approach is to omit the computation of LSVD
covariance. However,
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Table 16: Metries for models trained on ILI, with different length of sliding window. Each experi- ment is repeated by 10 times, and standard deviations are provided in brackets.
Model (ILI)
Window
CRPS
QICE
ProbCorr
Conditional FID
Tife
11
0.668(0.980)
8.341(2.465)
0.314(0.027)
5.715(1
2023
15
1.046(0.081)
13.597(1.550)
0.399(0.048))
6.845(0.813)
19
0.786(0. 120)
10.722(5.14)
0.345(0.064)
5.162(0.701)
SSSD
11
0.792(0. 1400)
10.807(1.84)
0.376(0.045)
5.282(0.675)
2023)
15
0.758(0. 110)
9.115(2.030)
0.365(0.033)
5.964(1.895
19
0.785
9.228(1.89)
0.376(0.040)
5,452(1.367)
Diffusion
11
0.873(0.
4.230(1.403
0.336(0.037)
9.619(2.963)
-TS (024)
15
0.769 168)
8.883(1.84) (1840)
0.373(0.058)
5.969(1.215)
19
0.894(0.100)
5.504(1.291)
0.327(0.013)
8.325(2.539)
TMDM
11
0.689(0.013)
7.081(2.059)
0.342(0.002)
6.567(2.991)
2024
15
0.722(0.025)
8.029(2.734)
0.359
12.234(18.767)
19
0.717(0.008)
7.297(1.610)
0.355(0.000)
7.954(5.131)
NsDiff 2025
11
0.646(0.086)
5.802(1.316)
0.344(0.034)
12.466(5.457)
15
0.645(0.059)
6.173(0.970)
0.307(0.058)
14.852(3.843
19
0.706(0.045)
6.268(1.186)
0.364(0.029)
12.375(3.949)
FLoWTS
11
0.898(0. 106)
11.338(0.764)
0.425(0.069)
6.713(0.998)
2025
15
(0.50))
10.645(0.778)
0.410(0.021)
6.202(0.536)
19
0.838(0.053)
10.721(0.865)
0.409(0.017)
6.391(0.951)
Metrics for DSPD and TsFlow trained ETTh1. Each experiment is repeated by 10 times, and standard deviations are provided in brackets.
Table 17:
Model (ETTh1)
CRPS
QICE
ProbCorr
Conditional FID
DSPD 2023)
0.741(0.090) 0.568(0.040)
11.032(0.816)
0.288(0.039) 0.257(0.026)
(9.28899554))
TsFlow 2025
7.968(1.018)
18.596(12.677)
DSPD
TsFlow
7.5
0.0
2.5
2.5
3.0
-5.0
-7.5
7.5
-1000
-10.0
25
75
100
125
160
175
90
75
100
125
175
CW-SSSD
CW-Flow TS
1.5
5.0
-3.5
-2.
5.0
5.0
-75
-7.5
-6.0
-10.0
75
100
125
150
175
50
100
125
175
Figure 5: Comparison of DSPD, CW-SSSD, TsFlow and CW-FlowTS on the first dimension of ETThl.
S In step (2) and (3), we by default compute c +0.5 using eigen-decomposition. However, this step in fact be avoided. Recall that the output of JMCE includes the conditional mean
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T and lower-triangular matrix Enee for t I....,,;; By directly computing the inverse of Ly c. we can whiten the time-series data without performing eigen-decomposition. We illustrate the idea by the following simple case. Suppose X R is random variable with covariance Cov(X) I.I" where is lower-triangular matrix. Then it straightforward to verify that
Cov(L L = LL
= lr-
Thus, we can replace (T in line 3 of Algorithm with Lnje- and similarly replace (I 10.5 line of Algorithm with Lnc. This transforms the eigen-decomposition step in the original algorithm into computing the inverse of a lower- twiarttra matrix. Since the inverse of lower-triangular matrix can be obtained efficiently using forward substitution, this modification yields a substantial speedup compared to performing eigen-decomposition (Strang (022).
Moreover, Weather and Solar Energy datasets, verify that this substitution substantially duces the computational cost, with the exact reduction reported in Table T9
E. CW-GEN AN END2END FASIIION
In the Section 3 and our training pipeline first trains JMCE, then conditionally whitens the time series data, and finally trains the generative model. However, with the accelerated algorithm intro- duced in Appendix E..9, we are able to train JMCE and the generative model jointly in an end-to-end (E2E) fashion, which further improves training efficiency.
In Table 18] we compare the setting without prior information (Raw), CW-Gen trained in the default manner (CW), and CW-Gen trained in an E2E fashion CCW-E2E). The results show that CW-E2E generally improves both ProbCorr and Conditional FID, while its QICE is slightly inferior to that of CW. The algorithm of training CW-E2E can be found in Algorithm 6 and compared the training time of CW-Gen and CW-E2E in Table
We also carefully
Table 18: Metrics for models trained ETTh1, including those trained data (Raw), the default CW-Gen pipeline (CW), and the end-to-end CW-Gen variant (CW-E22).. Each experiment is repeated 10 times, and standard deviations are provided in brackets. The best results underlined and the second-best results are -unhhed-udddrlied.
Model (ETThI)
Variant
CRPS
QICE
ProbCorr
Conditional FID
TifeDiff
Raw
0.787(0.051)
9.038(0.946)
0.320(0.012)
19.008(6.088)
2023)
CW
0.505(0.040)
8.821(1.916)
0.243(0.027)
6.788(5.425)
CW-E2E
0.514(0.039)
9.189(1.189)
0.218(0.016)
4.305(0.547)
SSSD 2023
Raw
0.836(0.153)
11.624(1.312)
0.326(0.032)
40.887(17.601)
CW
0.524(0.085) 4.838(1.921)
0.238(0.024)
9.265 .003)
CW-E2E
0.489(0.054)
6.254(1.612)
0.229(0.017)
6.908(3.625)
Diffusion -TS 2024
Raw
0.626(0.027)
3.002(0.838)
0.401(0.017)
81.563(60.905)
CW
0.445(0.024)
2.963(0.887)
0.266(0.012)
7,686(2.751)
CW-E2E
0.474(0.031)
5.536(1.514)
0.271(0.014)
5.105(1.
TMDM
Raw
0.472(0.031)
(360(1.055)
0.23010.014)
9.931 (4.439)
2024)
CW
0.440(0.001)
4.555(0.855)
0.213(0.001)
3.831
CW-E2E
0.433(0.027)
2.368(0.247)
0.205(0.010)
11.654(1.757)
NsDiff 2025
Raw
0.407(0.032)
1. 1.792(0.682)
0.214(0.014)
35.261(7.785)
CW
(.431(
(.249))
0.206(0.010)
8.820(1.541)
CW-E2E
0.412(0.010)
1.484(0.479)
0.195(0.006)
(1.27(1.264)
FlowTS
Raw
0.724(0.135)
8.820(2.631)
0.354(0.060)
39. 39.793(24.853)
2025
CW
0.488(0.020)
8.817(0.460)
0.254(0.021)
4.865 563)
CW-E2E
0.481(0.023)
10.277(0.455)
0.220(0.017)
4.040(0.741)
31

---

Published as a conference paper at ICLR 2026
Algorithm 6 Training JMCE and CW-Diff in an end-to-end fashion
Input: (No, C) in training set, hyperparameters Amin WEigens diffusion schedule 9r, [0, 1].
Output:
A trained JMCE model JMCE(.) and trained
neural network
CW
$8
1: Calculate sliding-windowd window covariances Ex..1...., Exo.ry of Xo
2: Initialize a non-autoregressive model J-CE(-) and neural network of diffusion model sg
3: while not converge do
4: Calculate Fxic. [ijc.... LTylC JMCE(C)
5: for 1= 1,... Ty do
6: Let Ex,,'/c -L,ccTu
7: Perform cigen-decomposition of Exo.tcc and obtain eigenvalues X
de. =1,11,11 ,,...,d
8: Perform singular value decomposition (SVD) of Σx,.t Exx,ofc and obtain singular values
85,,1 I,...,d
9: end for
Calculate L2 = ||x. -xxccl L = Ei' Exo,t Exo.1c//1 LSVD End E 81,1
10:
RA -ET11 Ei=1 ReLLU(mmn - -PExonicii)
Calculate LIMCE L2 + LSVD + Amiu v TyLF p Wrigen Rx
Calculate Le¹ [ 10.....
..... [T/]c
13:
Calculate X
L (Xo - (x|c) (Xo-
15:
Draw U10,
16:
Draw E N(0,
17:
Calculate a, = cxp { Jo 8,ds/2 ] and =²=11 1- a?
18:
Calculate LpaI || (a,X0 + t,c, C, c/a-11
CW
19:
Calculate LE2E = LINCE + Lpiff
20:
Calculate LEDE and update the parameters of JMCE(.) and sg
21:
end while
return JMCE(-), $p
F IMPLEMENTATION DETAILS
The evaluation setup, including the history length, prediction horizon, sliding window covariance, and the basic configuration of the JMCE loss, has been described in Section 5) In this section, we provide the detailed training parameters and implementation specifics for the proposed JMCE and the baseline methods.
For our JMCE model, except for the Solar Energy dataset, the backbone is Non-stationary Trans- former with a model dimension of dmodel = 512, attention heads, 2 encoder layers, I decoder layer, dropout rate 0. and feedforward layer dimension of 1024. For the Solar Energy dataset, dmodel is set to 128 and the number of encoder layers is increased to 3. Training is performed using the Hutter 2019) with learning rate of x 10-4, weight decay of 5 x 10-4, a batch size of 64, and 20 epochs. We select the model with the lowest loss over 20 epochs as the final model.
AdamW W optimizer (Loshchilov
For the baseline methods: the default parameters of SSSDDSA in (Alcaraz & Strodthoff 2023). Diffusion-Te TS uses the parameters for ETTh in (Yuan & Qiao| 2024). TMDM and NsDiff follow (Li al. 2024 Ye et al. 2025), with minor modifications to their own mean & variance estimators. FlowTS uses the parameters reported in (Hu et al. 2025). Except for TMDM and NsDiff, all other methods are trained using the AdamW optimizer (Loshchilov & Hutter 2019) with a learning rate of x 10-3 weight decay of 5 x 10-4, batch size of 128, and 20 epochs. We select the model with the lowest loss over 20 epochs the final model.
TifeDiff uses the default parameters in (Shen & Kwok 20233 SSSD uses
All of the experiments conducted single NVIDIA A6000, with memory of 48GB.
32

---

Published as a conference paper at ICLR 2026
Algorithm 7 Training JMCE and CW-Flow in end-to-end fashion
Input: (No, C) in training set, hyperparameters Amin Wigenn
Output:
A trained JMCE model JMCE(-) and
trained
neural
network
1: Calculate sliding- sliding-window window covariances Exp.1..... Xx.,T of Xo
2: Initialize non-autoregressive model JMCE(.) and neural network of flow matching "
3: while not converge do
4: Calculate Pxic, L Lic. Lic............ LT,iC i JMCE(C)
L.........
5: for = ,,...,,. do
6: Let E X,.'/C :.ajcc-.jja T.
7:
Perform aigendeccomposition of Exo.t(c and obtain eigenvalues A
8: Perform singular value decomposition (SVD) of Ex.,e Σxp.t/c and obtain singular values
Si,c. ,,...,d
9:
end for
10: Calculate 1,2 = 1|xo Txxcc II===
Exo.t Ex, ,
LsVD =ET1 Ei= 8i,
RA Et' Et= ReLU()min - -İSxo
11:
12:
Calculate LIMCE L2 + LIvD Amiu / TyLy Wrigeen RA
Let Lc = ..nc.......,,ee
13:
14:
Draw T 10
Draw eCW ~ ~N(0, LaxdxT;)
16:
Calculate e cCW = Lo C Pxxcc
17:
Calculate LFtOw e Xoo % (Xo + T(E -xo), r)l
18:
Calculate LEZE LIMCE + TrRow
19:
Calculate LEDE and update the parameters of JMCE(.) and e
20:
end while
21:
return JMCE(-), ofW
G COMPUTATIONAL EFFICIENCY
In CW-Diff, our algorithm first trains a JMCE model and then applies conditional whitening to each batch in the training The whitened batches subsequently fed into the diffusion model for training. This final stage requires essentially the same amount of time as a standard diffusion model; therefore, we refer readers prior work for details on training and sampling times Shen
& Kwok 2023 Alcaraz & Strodthoff 20233 Yuan & Qiao 2024 Li et al., 2024 Ye et al.) 2025] Hu et al. 2025). In CW-Flow, however, additional multiplications and additions on white noise are performed in each epoch, leading to extra computational overhead, as shown in line 8 of Algorithm Conscquently, the computation time we report includes the training time of JMCE, the time for conditionally whitening all batches, and the extra training time of CW-Flow.
The variability in computation time negligible, so we report results from a single run. The training time of JMCE primarily depends the dimensionality and length of the dataset, while the cost of conditional whitening is also affected by dimensionality. Moreover, since we use a highly parallel degemddecompoiitionn algorithm. the speed depends the number of batches rather than the number of samples per batch. Table 19 summarizes the training times on ETTh1, ETTh2, II.I, Weather, and Solar Energy. Because ETThi and ETTh2 have identical dimensionality and length, their computa- tional efficiency is the same.
For CW-Flow, the tensor operation in line of Algorithm 4 must be performed in every epoch. Its computational complexity is OddT?), which is not negligible cost. Fortunately, with advances in modern hardware and code packages, this operation can be executed a highly parallelized manner. The extra time is reported in Table 20
33

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Published as a conference paper at ICLR 2026
The dimensions, total length, the training time of JMCE (Train JMCE), the time of condi- tionally whiten all batches by eigen decomposition (CW eigen) and the time of conditionally whiten all batches by calculate the inverse of triangle matrix Lye (CW trig), the time of training a NsDiff model (Train NsDiff), and the time of training a CW-NsDiff in an E2E style (CW-NsDiff- Diss-EEE22E. All time are counted in second.
Table 19:
CW-NsDiff EEEE
212.9
212.9
157.2
1110
8120
Train NsDiff
482.92
684.45
79.8
79.8 9
7.1
CW trig
2.5
2.5
0.5
11.4
52.7
CW eigen
14.2
14460
2.8
2.8
0.6
Train JMCE
24185
156
156
58
780
Total length
14,400
(4,400
$2,696
52,560
966
Dimension
7
7
7
21
137
Solar Energy
Dataset
E
BTTh2
Weather
ILI
Overall, for datasets with low to medium dimensionality. CW-Gen remains highly efficient. How- ever, for high-dimensional datasets (such Solar Energy), CW-Gen becomes slower, since it requires performing numerous matrix eigen-decompositions, whose computational complexity is (dd))
34

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Published as a conference paper at ICLR 2026
Table 20: The dimensions, total length, the time of training a FlowTS model (FlowTS), and time of training a CW-Flow model (CW-FlowTS).
Dataset
Dimension
Total length
FlowTS
CW-FlowTS
ETTh1
7
14,400
147
152
ETTh2
7
14,400
147
152
ILI
7
966
11.4
12.2
Weather
21
52,696
720
740
Solar Energy
137
52,560
6580
10640
H THE USE OF LARGE LANGUAGE MODELS (LLMS)
In the process of preparation and writing of this paper, used CharGPT 5.0 the LLM tool for text polishing. The specific application scope includes optimizing the language expression of the abstract, introduction, experimental results, and discussion, improving the clarity and fluency of academic language, and adjusting the logical connection between sentences and paragraphs.
All content polished by the LLM has undergone strict review and manual editing ensure the accu- racy of academic concepts. the rigor of logical reasoning, and the originality of research conclusions. The authors bear full responsibility for the authenticity, integrity, and academic validity of the en- tire content of the article. The LLM tool was only used for auxiliary text optimization and did not participate in research ideas, experimental design, data analysis, or conclusion derivation, so it does not meet the authorship criteria.
35

---

Published as a conference paper at ICLR 2026
Metrics for CW-Gen models with different backbones of JMCE. Each experiment is re- peated by times, and standard deviations are provided brackets. The better results are under- lined. NS, FED, and IN indicate that the backbone of JMCE is the Non-stationary Transformer, FED-Former, and Informer, respectively. SEP indicate the mean estimator and covariance estimator separately trained, whose backhones are Non-stationary Transformer.
Table 21:
FED
(0.281)
22.970
(10.228)
32,600
(29.270)
31.392
(11.788)
18.389
(10.985)
32,906
(9.032)
16.7%
Conditional FID (L)
(5.415)
9.265
(5.003)
7,686
(2.751)
3831
(0.431)
8.820
(1.541)
4,865
(0.563)
83.3 39%
NS
(6.549)
21.523
(13,230)
15.609
(4.751)
12.139
(6.919)
15.080
(3.1233
8.686
(0 (9.996)
0.0%%
SEP
(0.416)
26,243
(15.228)
13.828
(13.082)
13.195
(4.345)
22.437
(7.024)
15.836
(3:045)
0.0%%
(0.014)
0.299
(0.024)
0.320
(0.037)
0.279
(0.046)
0.225
(0.010)
0.255
(0.015)
10. 10.7%
0.243
(0.027)
0,258
(0.030)
0.266
(0.012)
0.213
(0.001)
0.206
(0.010)
0.254
(0.021)
06 7%
ProbCort (
(0.020)
0.304
(0.055)
0.282
(0.016)
0.237
(0.022)
0.267
(0.011)
0.233
(0.010)
16.7%
SEP
(0.018)
0.273
(0.014)
0.279
(0.927)
0.251
(0:011)
0.263
(0.017)
0.249
(0.016)
0.0%
7.215
(0.859)
4.948
(2.927)
2919
(0.888)
(1.067)
1.432
(0.121)
4.088
(0.923)
50.0%
(1916)
4,338
(1 (1921)
2.963
(0.887)
(0.855)
1.249
(0.28))
8.817
(0.460)
33.3%
QICE (1)
(0.34)
8.120
(2.535)
8.615
(1.41)
(1.086)
7.653
(0.545)
5.200
(0.569)
0.0%
SEP
13.171
(1.712)
4,931
(1.953)
3.474
(1.430)
6.775
(0.737)
1,880
(0.272)
4,017
(0. 780)
16.7% 7%
0.372
(0.011)
0.543
(0.132)
0 1.385
00.015)
0.607
(0.049)
0.355
(0.005)
0. 18383
(0:069)
50 00%
(0.040)
0.524
(0.085)
0.445
(0.024)
0.440
(0.001)
0.431
(0.029)
0.488
(0.020)
33 3%
CRPS ())
(0.067)
0.714
(0.171)
0.652
(0.054)
0.527
(0.041)
0.640
(0.027)
0.491
(0.019)
0.0%
SEP
(0.032)
0.563
(0.098)
0.460
(0.029)
0.638
(0.023)
0.432
(0.014)
0.482
(0.017)
16. 7%
Diffusion TS
Model Backbone
TimeDiff
20233
SSSD
2023
2024
TMDM
2024
NaDif
2025
FloNTS
2025
Win rate
36


Reviews:
[{"review_id": "4pqEpzgYOC", "reviewer": "Reviewer_Nwt6", "summary": "The paper introduces CW-Gen, including CW-Diff and CW-Flow. It proposes a novel JMCE to simultaneously estimate the conditional mean and sliding-window covariance of future time series, which guides the whitening process. The authors provide theoretical guarantees showing conditions under which their approach improves the generative model's sample quality by reducing the KL divergence between the conditional distribution and the model's terminal distribution. Experimental evaluations demonstrate improvements in probabilistic forecasting accuracy, capturing non-stationary dynamics and inter-variable correlations better than baselines.", "questions": "See above", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper establishes conditions that justify why replacing the traditional terminal Gaussian distribution with one parameterized by estimated conditional mean and covariance improves sample quality.\n2. The JMCE simultaneously learns accurate conditional means and sliding-window covariances with eigenvalue control to ensure stability and robustness—this nuanced approach effectively addresses non-stationarity and heteroscedasticity.\n3. Experiments show the outperformed performance over baselines.\n4. The algorithms and theoretical proofs are clearly detailed, and code for experiment reproduction is available.", "weaknesses": "1. The approach involves computationally expensive operations, particularly eigen-decomposition for whitening covariance matrices, which scales cubically with dimensionality. For very high-dimensional datasets, CW-Gen can become quite slow, limiting real-time deployment scenarios.\n2. The framework's reliance on complex joint estimators and whitening transformations may impose a higher barrier to adoption compared to simpler baseline models.\n3. The paper ablates hyperparameters but does not isolate the individual contributions of conditional mean estimation versus covariance estimation. Which component drives most improvements?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JYKhvjX7IZ", "reviewer": "Reviewer_skbS", "summary": "The paper proposes CW-Gen, which whitens data using learned conditional means and covariances from a Joint Mean-Covariance Estimator before training generative models. The idea to align the model priors with data statistics is sound and yields consistent empirical gains across datasets.", "questions": "- Can you be very explicit on how your proposed method compare to other (univariate) flow-based and diffusion-based methods that introduced flexible priors, e.g., [1,2]\n- Can you add a short experiment showing numeric values of both sides of Equation (3) on training/validation examples so readers can see how far the estimators are from satisfying the bound in practice.\n- Choice of sliding window length (95 for most datasets, 15 for ILI) needs justification. Why 95? How sensitive are results to this hyperparameter?\n\n\n1. Modeling temporal data as continuous functions with stochastic process diffusion, ICLR 2023\n2. Flow Matching with Gaussian Process Priors for Probabilistic Time Series Forecasting, ICLR 2025", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Tackle a challenging, important and underrepresented aspect for time-series modelling: multivariate time-series.\n- Clear motivation: Proposes a method that allows for a new prior, that is closer to the data-distribution.\n- Well written and theoretically founded.\n- Extensive evaluation, which show that the proposed prior actually improves state-of-the art models.", "weaknesses": "- Unclear connection of DKL reduction in Theorem 1 to actual practical guarantees: Can you (a) clarify the tightness of the bound; (b) give intuition / toy examples showing when the condition is achievable (or not); (c) explicitly discuss regimes where the condition can fail\n- Unclear estimator quality: Covariance targets are noisy; stability and regularization not analyzed.\n- The paper notes that CW requires eigen-decomposition per sample / per time step but then glosses over practical limitations. More precise complexity analysis and discussion of remedies (approximate eigen/svd, diagonal + low-rank approximation, block-diagonalization, randomized SVD or factor models) could benefit the paper.\n- The paper argues joint mean+full covariance is important. But it’s not fully convincing which component drives gains. There are some ablations (backbone/wEigen) but I don’t see a simple controlled ablation that compares: (a) subtract mean only, (b) subtract mean and scale by diagonal variance (NsDiff style), (c) full covariance JMCE. Can you add a direct ablation and show where full covariance helps.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZvL7jyJNsd", "reviewer": "Reviewer_pWon", "summary": "The paper introduces conditionally whitened generative models that incorporate information in the noising process. The authors tailor diffusion and flow matching processes by including conditional information obtained via a mean and covariance estimator (termed JMCE). Their approach is theoretically motivated and empirically demonstrated to improve generative performance.", "questions": "- Can the model be trained in an end-to-end fashion or does it require a two-stage process?\n- Did you try diagonal covariance parametrizations to reduce the runtime complexity?\n- Can you elaborate more on the performance of the JMCE model itself? How does it compare in forecasting?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The methodology is theoretically justified, and the authors show under which conditions the distance between the prior and conditional distribution is minimized.\n- A simple mean and covariance estimator is introduced and well-motivated to parametrize the generative process.\n- The proposed conditional whitening leads to empirical performance improvements.\n- The framework includes diffusion and flow matching.", "weaknesses": "- The model requires a two-stage process now. First fit JMCE, then train the generative model.\n- The derived diffusion process requires the inversion of the covariance matrix, resulting in a higher runtime complexity compared to standard diffusion models. A runtime comparison would aid the comparison.\n- The limitations of the method should be discussed more thoroughly. Furthermore, I recommend separating the related work section from the introduction.\n\nMinors:\n\n- L260: Sentence incomplete", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aZuIPsrVj8", "reviewer": "Reviewer_wUr1", "summary": "The authors propose to use a separate model to predict sliding-window means and covariances and subsequently use those to whiten to prediction targets of generative forecasting models.\nTheir method improves forecasting results for a broad selection of generative forecasting models on several datasets.", "questions": "1. Do you train the JMCE model jointly with the generative model?\n1. Line 313: How can this be done more efficiently exactly?\n1. Are you modifying the generative models themselves? Or do you train the models on the conditionally whitened data as they are?\n1. Have you ablated the different components of your JMCE loss?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1. Consistent improvements in experimental results\n1. Clear graphical illustration of method", "weaknesses": "1. As far as I understand from the algorithms in the appendix, conditional whitening with a JMCE is a wrapper or pre/postprocessing around a generative forecasting model. While this can be interwoven with the diffusion/flow matching dynamics in Section 4, this does not seem essential. If I am correct, I think it would be an advantage to highlight the simplicity of the final method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)