DiffCATS: Causally Associated Time-Series Generation through Diffusion Models
Authors: Giuseppe Masi
Andrea Coletta
Elizabeth Fons
Svitlana Vyetrenko
Novella Bartolini
Venue: Accepted by TMLR ()

Abstract: Modeling and recovering causal relationships in time-series data can be crucial for supporting real-world interventions and decision-making, but progress in Time-Series Causal Discovery (TSCD) is often limited by the lack of high-quality datasets with diverse and realistic temporal causal relationships. This highlights the need to provide synthetic time-series generation tools, with realism as a primary objective, an aspect that requires incorporating causal relationships beyond mere correlation. To address this challenge, we propose a diffusion model called DiffCATS. It simultaneously generates multiple causally associated time-series as well as a ground truth causal graph that reflects their mutual temporal dependencies, requiring only observational time-series data for training. Experiments demonstrate that it outperforms state-of-the-art methods in producing realistic time-series with causal graphs that closely resemble those of real-world phenomena. We highlight the practical utility of our data on three downstream tasks, including benchmarking widely used TSCD algorithms.

Published in Transactions on Machine Learning Research (01/2026)
DiffCATS: Causally Associated Time-Series through Diffusion Models
Generation
Giuseppe Masi
Sagienza University of Rome
Andrea Coletta*
andrrn.colcctaaaaannnaaaaaa
Banca d d'Italia
Elizabeth Fons J.P. Morgan Al Research
elizabeth.fons. fomooornnmomooocommmmmmmm
Svitlana Vyetrenko
svitlana&&oussampier.com
Outsampler, Strusboury, France
Novella Bartolini
novella@di. unwromm....
Sagienza University of Rome
Reviewed on OpenReview: / net/ forum? id- PuC60yaHHop
Abstract
Modeling and recovering causal relationships in time-series data can be crucial for supporting real-world interventions and decision decision-making, making, but progress in Time-Series Causal Discovery (TSCD) is often limited by the lack of high-quality datasets with diverse and realistic temporal causal relationships. This highlights the need to provide synthetic time-series generation tools, with realism as a primary objective, an aspect that requires incorporating causal relationships beyond mere correlation. To address this challenge, we propose a diffusson model called DiffCATS. simultancously generates multiple causally associated time-series well ground truth causal graph that reflects their mutual temporal dependencies, requiring only observational time-series data for training. Experiments demonstrate that it outperforms state-of-the-ad methods in producing realistic time-series with causal graphs that closely resemble those of real-world phenomena. We highlight the practical utility of our data on three downstream tasks, including benchmarking widely used TSCD algorithms.
1 Introduction
Many real-world time-series can be usefully modeled as arising from directed (causal) relationships among variables, which motivates methods for representing and learning such structure from data (Runge) 2018 Runge et al. 2023). Understanding such causal relationships is well-recognized and important challenge for decision- and policy formulation, as it facilitates predicting the consequences of interventions on underlying systems and variables (Hasan et al. 2023
Over the years, several works have studied these underlying causal structures, starting from Granger Causality (GC) Granger 1969). Unable capture how time affects causal relationships between interdopendent
time-series, GC has been complemented by Causal Graphs that incorporate the temporal lag in which causality
unfolds [Pearl 2009). Many approaches tackling the Time-Series Causal Discovery (TSCD) problem (Hasan
ct al. 2023) achieve satisfactory performance using statistical and machine learning techniques (Runge
"The opinions expressed this paper personal, and should be attributed Banca d'Italia The research work was parried out before the author joined Banca d'aaa..

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Published in Transactions on Machine Learning Research (01/2026)
Imnen Lenogr
403-
Dilingen
252
202-
Kempten
Lenggries
30
15
20
25
30
steps
Figure 1: A synthetic sample and causal graph of three river discharges in which Kempten has an effect on Dillingen with lag of 1.
Cattr Geign
Figure 2: Two generated examples in which the causal graph helps a predictor model to reconstruct the target feature.
et al. 2019 Pamfil et al. 2020 Sun et al. 2023 Cheng et al. 2023), with discovered causal graphs closely resembling the ground-truth counterparts. However, the limited data available may hinder the development of new methooologie and studies, raising concerns about, how existing algorithms would perform in unseen real-world scenarios (Cheng et al. 2024
Novel methodologies to generate realistic datasets with rigorously defined causal graphs are needed to support research and development of algorithms on time-series causal graphs. This challenge has been recently lackled by the works Li et al. (2023) and Cheng et al. (2024), which mark an initial step in this direction, proposing two deep learning models to generate synthetic time-series data, while extracting the corresponding causal graphs. The first model focuses on the concept of Granger causality and proposes a recurrent Variational Autoencoder (CR-VAF) framework that naturally encodes causality into the weight matrix connecting input and hidden states. The second model introduces comprchensive framework that supports prior causal graphs to generate realistic lime-series data. However, when input causal graph is not provided, the method extracts a hypothesized causal graph using explainability tools for feature importance (e.g., DecpSHAP (Lundborg 2017)), which are inherently slow and only provide a postcrior approximation of the ground-truth graph.
In this paper, we introduce a novel generative framework called DiffCATS that combines the advantages of previous approaches by jointly generating time-series along with its causal graph, directly within diffusion model architecture. Specifically, our model incorporates r-lag vector autoregressive structure (VAR(r)) for multivariate time-series (Hamilton) 2020), where the coefficients are learned and generated through the diffusion process. This approach enables the simultaneous generation of realistic time-series data and the derivation of the corresponding ground-trun truth causal graphs from the VAR coefficients (Zivot & Wang 2006). DiffCATS can be trained directly on time-series data without requiring prior causal graphs, eliminating the need for additional explainability tools.
Our work facilitates research and development of efficient algorithms for uncovering cause-effect relationships in multivariate time-series across diverse fields. In particular, the generated synthetic data can complement real-world data by providing known ground-truth with diverse set of causal structures, which is especially valuable when real data are scarce or have limited ground-truth causal structures. Different from existing work, our approach specifically addresses the coherence between the synthetic sample and its corresponding causal graph (sce Figure 1], resulting in more realistic and uscful synthetic data. As the experiments highlight, the generated causal graphs are key understanding the underlying dynamics of time-scries, improving predicting capabilities (see Figure
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The main contributions of this work are the following:
We present DiffCATS, novel pipeline that employs diffusion model to generate realistic time-series along with their related causal graphs.
With extensive experiments, we demonstrate that our method outperforms existing approaches, terms of synthetic time-series quality and fidelity of causal graphs to real-world phenomena.
We conduct am evaluation of existing causal discovery algorithms using our synthetically generated datasets, highlighting the practical benefits of our data.
We demonstrate the utility of the causal graph and its coherence with time-series data through two additional downstream tasks.
2 Related work
Several works have addressed the generation of synthetic time-series starting from real datasets (Yoon et al.
2019: Jarrett et al. 2021 Rasul et al. 2021). Some approaches have focused on specific aspects, such as the
correlation dynamics among variables (Seyfi et al. 2022 Masi et al. 2023), user-specified constraints (Coletta
let al. 2023), or interpretable generation methods (Yuan & Qiao| 2024; Fons et al. 2024p. However, only a
few works delve into the generation of time-scrics along with their causal structure (Li al. 2023
Chong
et al. 2024).
The work of Li et al. 2023 proposed VAE-based framework capable of learning Granger causal relationships from real multivariate time-scries. This approach derives causal relationships from the weight matrix of the model connecting the input and hidden states, learning a unique Granger causality matrix from the data to which all generated samples adhere, A recent work of Cheng et al. (2024) proposed a pipeline generate rcalistic time-scries along with the causal graph. However, their framework does not output an seeepretableby-ddssiee time-serics, but it performs the hypothetical causal graph inference through DeepSHAP (Sundararajan & Najmi 2020) on the trained generative model, introducing a considerable time overhead.
Our goal is to further explore this area and address the gaps in the current literature. Specifically, we aim to provide a novel generative approach to simultaneously generate multiple causally associated time-series and their causal graphs, incorporating temporal lags. We strive to generate a unique causal graph for each synthetic sample, introducing greater variety in the data and providing a naturally interpretable architecture.
3
Problem Formulation
3.1 Background Knowledge
Causal Discovery The Causal Discovery task aims to identify cause-effect relationships among the variables of a d-variate time-series = (xx ......"."" x²). We say that x' has an effect on (or causes) x if the two variables reflect real phenomenon in which events reflected in the values of xt affect x². Trivially, the cause must precede the effect, so it is important to consider also the lag that clapses between obscrving the cause event on x' and the effect event on x. Causal Discovery algorithms are employed to observe real data and point out the existence of causal relationships according to which xf causes x' after T time-steps, returning (x', x T).
Causal Graphs Causal relationships are often represented in the form of Causal Graphs. Let Tmaz E N+ be the maximum number of discrete time-steps (ôt) we are interested in to model the cause-effect phenomena of a. We define a Causal Graph G = (V, F)) where the vertices V represent the time-series variables for the various time-steps between - -maa and 0, and the edges E represent their causal relationships. In particular, edge (r/, xx{)) E E indicates that the variable x² has a causal implication on the value of the variable with a lag of l2 - f time-steps (i.e., ot, = r Formally, V (x|_1 0 < i S d, 0 < 7 S Tmazh and F. ((x), + xt with lag of la 2 0}- Figure illustrates causal graph describing the
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Synthetic Time-Series
NO... r)
Diftusion Model
Causal Graph
Cousal Coefficients
C2f0
Figure 3: DiffCATS pipeline. Given a noisy sample z ~ [O 1) the diffussin model runs the denoising steps and outputs (i) an initial prefix of Tmaz steps of the multivariate Lime-series and (ii) causal/VAR coefficients (C)) Fmar, whose entries *f parameterize directed influences from variable 81 lag r to variable ig time I. For each >> Tmax: the remaining samples are produced via causal reconstruction (cirele "x ic., VAR-style update using the previously reconstructed window and the generated matrices (including Co(!)). This yields the full synthetic time-series (solid prefix with dashed continuation). A sample-specific causal graph is then derived from the generated coefficients by retaining the most significant relationships, with nodes representing variables across time lags (c.g., 22.10
interdependencies among rivers according to the observations made by Ahmad et al. (0022). shows that variations in the water level of one river affect the level of the other onc.
3.2 Task Definition
Let. D = {æ x E [2dd be a set of d-dimensional inpul time-series of length L. Our goal is to use the data in D to train a generative model that best approximates the distribution of the time-series, while simultancously learning the corresponding causal structures. In particular, we aim at generating couples (0, where & Lxd is synthetic time-series and å is causal graph. We want & be similar to the time-series observed in D, i.e., to show realistic behavior that reflects similar statistical properties. We also require & to reproduce realistic causal relationships, observed in D, and described by the causal graph ğ that will explain è in terms of causal relationships. We formally define causal relationship between two time-series variables in the following section (Definition) T.
4 Methodology
Our framework, illustrated in Figure 3 builds upon powerful class of generative models, the diffusion models (Ho et al.] (020). Originally designed to generate realistic images, diffusion models are used in this paper to generale synthetic samples (æt, g). Unless otherwise noted, we adopt two common assumptions of the Causal Discovery literature (Cheng et al. 2024 Runge et al., 2019 Pamfil et al. 2020; Sun et al. 2023 Markovian conditions and faithfulness, as discussed detail in Appendix A.1 While these assumptions are crucial for causal discovery, the key consideration for generation from observational data ensuring structural consistency between the synthetic time-series and the causal graph. Indeed, unlike previous works, do not need to assume stationarity of the underlying causal phenomena, as our causal graphs will be strictly related to individual samples.
4.1 Diffusion framework
A diffusion model is type of latent variable model that operates through two key processes: the forward process and the reverse process. Given sample 2o E T the forward process gradually adds Gaussian noise
Although the proposed methodology is general and can, in principle, be applied to alterattve generative paradigms such VAEs Kingmaa & Welling (2013) GAN Goodfellow ct al. (2014), these models often underperform diffusion models due restrictive assumptions and less stable training dynamics. Consequently, focus Liffusion-based architecture, which
carefully design for our setting.
2In this section, the subscript refers the diffusion step, not the lag index used elsewhere.
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The architecture of DEN, consists of an initial convolutional layer followed by a series of RESNET and Published in Transactions on Machine Learning Research (01/2026)
to obtain noisy sample ør (O.. Specifically, given the parameters 8r (0.1) to schedule the amount of noise added at diffusion step / E [] the noisy sample is given by æ, V ao v1 â e where ~~ (0, = 1-11- and â; = II.
The reverse process performs the actual generation of a new sample starting from Gaussian noise. Following the formulation of Ho al. (2020). we perform the denoising procedure from ar ~ NO, according to the following equation:
21-1 = 3r V 01-1 ā,
2o + (1 àr-) .va 1 - âe
1- ---1 E. 2t + 1 [t20) B
(1)
1 - ái
In the above equation (c) is the indicator function, ~ (o,,,),, and t. DENo( t) is the output of neural network DEN, parametrized by 0, trained with respect the following loss function: ( lo f) ||æ0o - 2o where I1-1lp indicates the ky-norm. In practice, DEN, reconstructs the original sample taken from the dataset by filtering out the noise added during the forward process.
In addition to the fo-norm, we also consider other loss functions to improve the performance of the recon- struction, such as the Dynamie Time Warping-based term LDTW (wro &o; 0) introduced by Cuturi & Blondel 2017). The training objective is:
L(xo, to; 0) = TyCCeeeeeeeeeeeeeee to;0) Y (DT ao; 0)},
(2)
2n~D
where À is parameter weighting the additional term.
ATTENTION blocks (see Appendix B.3 for more details).
4.2 Causal reconstruction of the time-series
This section details how the output process inherently embeds a causal structure, allowing for the generation of a coherent sample (fo.g). Given 2o E 'D we denote with @a(() the value of the time-series at time I, for I E [1, L), and considering its components as distinct features, we denote with (l) the value the i-th feature at time 1, for E d1.
Let Tmaz E N+ be the maximum lag for the causal relationships in the synthetic time-series Simultaneously for each feature DEN, outputs the first Tmax steps of the time-series, i.e., (1), V1 S S Tmaz, and a set of causal matrices C(I) = ={c+()) = (c)(=()) ISsnisa E Rdxd] 0 S r S Tmaz] for all the remaining steps Tmax < <1 I,, where (1) represents the impact of feature i on is with lag of T at time I. The reconstruction of the whole time-serics according to causal relationships evolves through Vector Autoregressive (VAR) model (Zivot & Wang 2006): proceeding one step 1 at a time with Tmar <1s L,
-Tm Tmur C() a(1-7) -
=
We underline that, even though the reconstruction can be described by a VAR model, the generation framework is not autoregressive. This is because the model does not consider previously generated outputs inputs. It instead generates the initial time-steps and the coefficients simultaneously.
Finally, to encourage the model to focus on the most important causal relationships, we add a regularization term for the coefficients. While the ideal choice for such function would be the 4--nomm this is difficult
to optimize, therefore we consider the fz-norm, as in (Sun et al. 2023 Li et al. 2023). Specifically, the regularization is defined as:
Σ Rie (Zo; d2 Li
Tmaz (1)22),
(3)
==0
where da the weight associated to such regularization term.
The maximum lag and time-step gramularity can be decided according to the domain and expert knowledge.
Note that the matrix modeling instantaneous relationships Co(l). VI is generated lower triangular matrix with the elements the main diagonal also agreement, with Hyvärinen al. (2010).
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4.3 Causal Graph Extraction
While the broader field does not ofler single universally accepted characterization of causality in observational time-series, we will provide a flexible definition based on predictive formulation that is both widely adopted in practico and grounded in robust thcoretical principles. Indcod, given a synthotic sample &g reconstructed through the series of coefficients matrices <(l) it means that for each time-step Tmaz I L, for each feature i S d, we have importance weights (i)),.... (()....,,TT.... CTmo. assigned to the previously generated time-steps, i.e. the window [@o(l - Tmae),..... fo(l 1)] and instantaneous relationships in &o(() according to Co((). We also call these coefficients the explonation of the synthetic sample. To infer the causal graph 9, we summarize the causal relationships from the VAR coefficients according to the following formal definition.
Let p be the percentage of causal relationships we want to represent in the synthetic dataset. For synthetic sample ', we say that a' (p.q)-come -cauuccssess a) with lag of T if the q-quantile of the corresponding coefficients of the VAR model al log T is among the p% highest absolule volues. Notice that p and q refer to the whole dataset and the single sample, respectively.
Definition 1
Intuitively, the g-quantile value allows the identification of significant causal events within sample (c.g., Figure 1 by aggregating the coefficients over the time-series. Additionally, the threshold p can be employed as reference point to identify and constrain causality to focus on the most relevant causal relationships. The whole mechanism allows samples to have different causal graphs, i.e., some may exhibit dense connections while others may have none. In fact, unlike previous work, we do not assume stationarity of the causal relationships the dataset: the causal graphs are strictly related to individual samples, enabling more realistic and diverse samples.
This approach enables fair evaluation of TSCD algorithms (see Section 6.1 Specifically, TSCD algorithm may fail to recover the ground-truth causal graph derived from our prior knowledge of the system if the given sample does not exhibit any causal effect but is still linked to that prior causal graph. Our dataset serves as an accurale and representalive benchmark: each generated sample is explicitly linked to its corresponding causal graph, which may denote the absence of causal relationships when none emerge from the time-series.
Finally, we notice that our definition ensures that each identified causal link captures not only statistical predictability but also meaningful, information-tteoreliic causal effect. In fact, our formulation is consistent. with prior work that extends Granger causality with lag-specific formulations (e.g., (Hyvärinen et al. 2010)). as significant predictive link also implies high transfer entropy from the cause to the effect (Runge et al. 2012), indicating a substantial information flow between two variables. This information flow is the type of relationship captured by our definition of causal links as discussed in more details in Appendix A.2
5 Experiments
In the experiments section, we show that the proposed pipcline is able to generate high-quality synthetic samples along with coherent and realistic causal graphs. In this regard, we conducted an experimental campaign involving three different datasets. We compared our model against several state-of-the-art approaches to highlight its advantages. We evaluate the generated samples both quantitatively and qualitatively, using well-cstablished metrics for syuthetic time-series well as metries specifically designed to assess the realism of the causal graphs. The code to reproduce the experiments is publicly released 3
5.1 Datasets
To evaluate the models' capability to generate time-series alongside their causal relationships, utilize two real-world datasets (Rivers and AQI) and a synthetic dataset (Hénon) constructed using closed-form equations, for which we have ground-truth knowledge. The datasets are described in the following paragraphs while we refer the reader to Appendix B.1 for additional details, including the visualization of the ground-truth causal graphs.
Rivers: introducod by Ahmad et al. 2022 it consists of the average daily (St day) discharges of the
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Iller River at Kempten, the Danube River at Dillingen, and the Isar River at Lenggries between the year 2017 and 2019. The Melrr is a tributary of the Damube, and we expect that an increase in the water level of the former will flow into the latter within a day, ie., with a lag of 1 time-step. In this case, d = 3 and the only causal relationship is x 2x- + x, Dillingenn
Air Quality Index (AQI): introduced by Cheng et al. (2024), it consists of the PM2.5 pollution index monitored hourly (ôt = 1 hour) over the course of one year by 36 stations spread across Chinese cities. In this case, d = 36 and the available causal relationships are modeled through Granger Causality matrix, which is based on the pairwise distances between sensors.
Hénon: introduced by Li et al. (2023), this synthetic dataset consists of d = 6 coupled Hénon chaotic maps (Kggiutzii 2013) in which there is one positive (ri-2 + xf) and two negative causal relationships
(-x²-1 x and 1/-1 x ) A1 < i S d. The equations generating this dataset are described
in Appendix B.1
In the experiments, the maximum lag Tmaz: introduced in Section 4.2 is fixed to 2 for all the datasets, consistent with the maximum lag observed in the ground truth. A sensitivity analysis, and a discussion, of parameter Tmar is provided in Appendix C.3 The sequence length is 32 for the Hénon and the Rivers datasets, and 24 for the AQI dataset.
5.2 Models
Benchmarks.
We compare our model against the two most recent state-of-the-art works described in Sec-
tion 2 namely CAUSALTIME (Cheng et al. 2024 and CR-VAE (Li al. 2023). We also include two
other bascline solutions using diffusion models: (i) BASE-DIFFUSION
vanilla diffusion framework applied to time-series generation; (ii) CSDI (Tashiro et al. 2021). appropriately adapted to a generalion lask as described by Coletta et al. (202)). A detailed description of these models can be found in Appendix B.2
DiffCATS: We trained our model setting the parameters in Equation (2) and Equation (3) to X = 0.01
and A2- = 1, respectively. An extensive ablation study about the impact of each loss component along with
additional loss functions, namely the f--norm and Fourier-based loss, is presented in Appendix C.2 The
full list of hyper-parameters shown in the Appendix in Table 7 To extract the causal graph from the VAR coefficients, we set p 1% and 9 = 0.95 (see Definition
5.3 Evaluation Metrics
Evaluation of time-series. We evaluated the quality of the synthetic time-series using the following metrics for fidelity, usefulness, and diversity: DISCRIMINATIVE ScORE (Discr.) Yoon et al. 2019), PRE- DICTIVE SCORE (Pred.) (Yoon et al. 2019). AUTHENTICITY (Auth.) (Alaa ct al. 2022), MAXIMUM MEAN (xCorr.). All of them are well
DISCREPANCY (MMD) Gretton et al. 2006) and TROSS-CCORELLA
described in Appendix B.4
Evaluation of Causal Graphs. We note that, despite the existence of causal phenomenon relating the variables of the datasets, not all the samples extracted from the long time-series may exhibit clear evidence of this. For instance, concerning the Rivers dataset, even if the Iller is a tributary of the Danube, if there is no significant variation in the water level of the former, the phenomenon of causality cannot be observed. Indeed, the water level of the three rivers simply remains stable over substantial periods of time. As a result, many time-step windows extracted from the dataset will not provide evidence of the causal ionship. Considering this issuc, we employ metrics that do not penalize missing causal relationships from a sample. In contrast, we want to penalize those causal relationships identified by the model when domain knowledge clearly indicates that such relationships do not exist in real data (for instance, tributary cannot be caused by the recipient river). The recent works of Ahmad et al. (2022) and Hasan et al. (2023) addressed this problem by introducing metrics based on the false positive rate of causal relationships.
Accordingly, we consider the GRANGER CAUSALITY FALSE POSITIVE RATE (GC-FPR) and the CAUSAL GRAPH FALSE POSITIVE RATE (Graph-FPR), which account for the fraction of edges the graph known to be incorrect. Notice that, since CR-VAE does not output a causal graph for each sample, we use the F1-SCORE to evaluate its causal relationships with respect to the ground-truth Granger Causality matrix.
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We emphasize that the generative method proposed in this paper is not designed to be used causal discovery algorithm, so we do not focus on the exact causal relationships expected to be extrapolated from the datasets. Rather than that, we focus on the realism of the causal graphs and their consistency with the corresponding generated synthetic time-scries, ensured by the design of the generative pipeline.
Finally, we evaluated the INFERENCE TIME (Inf. time) of the models, i.e. the time to generate a synthetic sample and the corresponding graph.
5.4 Results
In this section, we discuss the results of our experiments. All the quantitative scores are shown in Table We report the results of DiffCATS with respect to the state-of-Che-arl approaches discussed in Section 5.2 namely, BASE-DIFFUSEE and CSDI for comparing the generated time-series, and CAUSALTIME and CR- VAE for comparing both time-series and causal graphs. We point out that Base Diffusion and CSD1 are included as reference points, to show that DiffCATS time-series quality is not far from high-performing time-series-only generators, while additionally providing causal graphs. All the results report the mean and standard deviation across 10 different seeds.
Table 1: Results of the models on the three datasets, where indicates lower is better and indicates higher is better. For each metric, the best result is highlighted in bold, and the second-best result is underlined (restricted to models able to generate both time-series and causal graphs).
Models Teme-Series Only
Models Time-Series & Causal-Graphs
Dataset
Metric
BASE DIPPUSION
CSDI
DIFICATS
CAUSAL
CR-VAE
Discr. 4
0.032 = 0.008
0.026 0.011
0.032 0.017
0.311 0.142
0.243 = 0.113
Pred.
0.170 0.005
0.221
0.006
0.156
0.009
0.200 0.011
0,211 0.012
Auth. f
0.679 0.005
0.905
+ 0.015
0.693 0.008
0.720
0.031
0.651 = 0.1
Hénon
MMD
0.001
0.000
0.001
0.000
0.001 0.000
0.002 0.000
0.012 0.009
xCorr
0.034 = 0.005
0.005 0.094
0.029 0.005
0.060 0.022
0.131
0.031
GC-FPR L
0.311 0,001
0.482 0.041
0.520 + 0.070'
Graph-FPR 4
0.017 = 0.000
0.231 0.010
Inf. time 4
1231ms
241ms
1548ms
8790ms
194ms
Diser. 4
0.074 = 0.007
0.019 + 0.022
0.067 = 0.010
0.090
0.050
0.110 = 0.090
Pred. J
0.030 0.001
0.043
0.002
0.033 0.001
0.026
0.001
0.036
0.002
Auth.
0.586 0.008
1.000
+ 0.000
0.630 0.010
0.560
0.030
0.720
0.020
Rivers
MMD
0.001 0.000
0.002
1 0.000
0.001 0.000
0.009
0.011
0.059
0.029
xCoir J
0.014 = 0.004
0.006 0.018
0.020 0.010
0.010
0.000
0.120
0.020
GC-PPR &
0.000
0.570
0.010
1 0.140"
Graph-FPR
0.000 0.070
0.220
0.010
Inf. time 4
1252ms
230ms
1492ms
4248ms
148ms"
Discr.
0.287 0.010
0.238 1 0.068
0.293 i 0.050
0.460
0.020
0.250 1 0.040
Pred. J
0.043 = 0.001
0.048
1: 0.001
0.047 = 0.001
0.054 = 0.001
0.043 : 0.001
Auth.
0.860
0.008
0.980
0.013
0.820 0.010
0.770
0.010
0.800 0.100
AQU
MMD
0.001 0.000
0.018
+ 0.001
0.001 0.000
0.008
0.001
0.017 0.001
xCorr
0.078 0.005
0.080 + 0.003
0.070 = 0.010
0.030
0.010
0.120 = 0.010
GC-FPR 1
0.390 0.000
0.490 0.000
0.270 = 0.000*
Graph-FPR.
Inf. time 1
1079ms
242ms
1395ms
205s
442ms"
Considering only the models that are able to generate both the time-series and the corresponding causal graph, we can see that regarding the fidelity and the quality of the synthetic time-series, DiffCATS outperforms the other approaches in terms of MMD on all three datasets, maintaining satisfactory degree of AUTHENTICITY. It is also the best model concerning the DISCRIMINATIVE ScORE on two out of three datasets, and in all the other cases, it obtains scores very close to the benchmark. This validates our generated samples with respect to their originality, usefulness, and indistinguishabiitty from real data.
The results comparing DiffCATS' time-series with those generated by the two state-of-theear models capable of producing only time-scries demonstrate a comparable level of quality between them. This ensures that the generation of the time-series through the VAR coefficients does not sacrifice the ability of the model to produce high-quality samples.
Regarding the causal graphs, DiffCATS achieves both the best GC-FPR and the best Graph-FPR scores all three datasets, demonstrating its effectiveness in extracting causal relationships from the VAR coefficients. This result is of critical importance given that it ensures the reliability of the graphs as representation
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of the causal relationships exhibited by the time-series. For the AQI dataset, report only the GC-FPR metric that evaluates the Granger Causality matrix, as no lag information is provided in the ground-truth causal phenomena. We recall that CR-VAE does not output a causal matrix for each sample, but it is learned and fixed in the trained model. Therefore, since the Granger causality matrix is the same for all generated samples, the GC-FPR metric does not fully capture the model's ability this context. For this reason we reported the F1-score of the learned matrix with respect to the ground-truth Granger causality matrix, highlighting room for improving performance.
Summarizing, highlight that our model achieves the lowest DISCRIMINATIVE SCORE and MMD along with the bost Graph-FPR, ensuring that the synthetic samples exhibit high level of realness and the causal graphs are reliable. Furthermore, as evidenced in Figure [] the model is able to generate synthetic traces with strictly associated causal graphs, eliminating the need for stationarity assumptions in causal relationships employed by previous works.
We also evaluated the inference time of the models to obtain sample made up of the synthetic time-series and the corresponding causal graph. CR-VAE turned out to be the fastest, thanks to its VAE-based architecture. However, great time saving occurs because the causal graph is fixed for each sample since it is extracted from the parameters of the model. Our model achieves an inference time significantly lower than CAUSALTIME, mainly because the post-processing of the feature importance through DeepSHAP is very time-consuming. Instead, in our architecture, the causal graph is generated simultaneously with the time-series, with only a moderate overhead. Moreover, the sampling of diffuusion models can be accelerated, using, for example, implicit diffusion models (Song et al. 2021).
Additional experiments and results can be found in the Appendíx, including the evaluation of the timo-scries through dimensionality reduction techniques, namely I-SNE and PCA C.4]. kernel density estimation C.5
the evolution of the evaluation metrics during the training [.6). , the robustness of the graph extraction to noise in the time-series C.7.1 and a different approach to extract the cansal graph from the VAR coefficients
using the Dixon's Q Test C.7.2
6 Downstream Tasks
6.1 Benchmark of Causal Discovery Algorithms
Our primary downstream task is to benchmark causal discovery algorithms. Given generated couple (), we feed the algorithm with the generated lime-series & and we compare the predicted causal graph against. the generated graph g. Related work presenting this type of benchmark is described in Appendix E.1 In our benchmark, we included the following approaches:
Granger-Causality based: Granger Causality (GC, (Granger 1969 ); Neural Granger Causality
(NGC, (Tank et al.) (02T)); economy-SRR (eSRU, (Kanna & Tan 2019 Temporal Causal Discov-
ery Framework (TCDF, (Nauta et al.) 2019 CUTS Cheng et al., 2022); CUTS+ (Cheng al. 2023);
LiNGAM-basson based: ICA-LINGAM (Shimizu et al.)
2006); VARLiNGAM (Hvärinen et al.
2010 Di-
recLLiNGAM (hhiiiuu et al.
2011);
Constraint- based: PCMCI+ (Runge et al.
2020
Gradient-based NTS-NOTEARS (Sun et al.
2023
DYNOTEARS Pammil et al. 2020 Rhino GGog
et al. 2023); :
based: Latent Convergent Cross Mapping (LCCM, (De Brouwer et al. 2020
Other: Neural Graphical Model (NGM, (Bellot al. 2021)) employing neural ordinary differential equations.
The results are shown in Table 2 evaluated in terms of AUROC and AUPRO (Area Under Procision-Recall Curve). To always have a well-defined ground-truth, for this benchmark, we selected the strongest 15% causal connections for each sample. As additional experiments, we also executed the benchmark using the top 1%
approach described in Section 4.3 These results are reported in Appendix E.3
The comparison between real and synthetic data performance demonstrates encouraging coherence across multiple causal discovery algorithms, suggesting that the synthetic data effectively captures key characteristics
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of real-world causal relationships. Several high-performing methods, including PCMCI+, CUTS, and CUTS+ show remarkably consistent performance between real and synthetic datasets, with AUROC differences typically within 0.05, indicating that the synthetic generation process successfully preserves the essential causal structure and complexity. The synthotic data proves particularly effective for methods like NGM and eSRU, which achieve consistently strong AUPRC scores (0.73-0.81) across both real and synthetic versions, demonstrating the synthetic data's ability to maintain predictive relationships. Notably, the AQU dataset shows the strongest real-synthetic correspondence across methods, while the synthetic Rivers dataset. successfully replicates the challenging characteristics of its real counterpart. While some methods, such NTS-NOTEARS and TCDF, exhibit performance variations, these discrepancies appear more attributable to algorithmic sensitivity rather than fundamental limitations in synthetic data quality. Overall, these results indicate that the synthetic data provides a high-quality approximation of real-workd causal dynamics, offering a valuable benchmark that captures the essential complexity needed for robust causal discovery evaluation.
Table 2: Results of the Causal Discovery algorithms on real and synthetic data. For real data, the highest value is shown in bold and the second-highs value is underlined. For synthetic data, green background indicates comparable performance to real data, while red background indicates high difference from real data performance.
Real Data
Synthetie Data Rivers
Metric
Method
Hénon
Rivers
AQI
Hénon
AQI
GC
0.53 +
0.52 0.03
0.57 + 0.07
0.00
DYNOTEARS
0.62 0.11
0.54
0.49 0.03
0.60 + 0.04
0.51 + 0.03
0.00
NTS-NOTEARS
0.48 0.00
0.50 0.00
0.50 0.01
0.57 + 0.04
0.69 + 0.10
+ 0.00
PCMCI+
0.70 1 0.14
0.71 1 0.28
0.72 0.04
0.74 1 0.02
0.77 1 0.06
0.74 0.00
Rhino
0.53 = 0.01
0.62
0.51 + 0.01
0.53 + 0.06
0.50 + 0.00
CUTS
0.80 = 0.10
0 0.01
0.78
0.75 + 0.02
0.76 + 0.06
0.71 0.00
AUROC
CUTS+
0.76 + 0.01
0.72
0.79 0.00
0.75 + 0.02
0.75 4: 0.08
0.74 +: 0.00
Neural-GO
0.68 = 0.01
0.50
0.50 0.00
0.72 + 0.01
0.52 0.05
0.50 + 0.01
NGM
0.60 + 0.03
0.68
0.03
0.61 + 0.06
0.69 0.12
0.50 0.00
LCCM
0.54 1 0.02
0.00
0.00
0.55 1 0.00
0.50 0.00
0.52 0.00
<SRU
0.50
0.78
0.50
0.50
0.75 0.11
0.00
TCDF
0.00
0.00
0.52 + 0.03
0.01
0.00
ICA-LINGAM
0.06
0.22
0.52
0.44 1
1 0.29
VARLINGAM
0.50 0.06
0.56
0.59
0.50 + 0.06
0.62 + 0.31
0.48 0.03
DirecLINGAM
0.51 + 0.06
0.72 27 0.29 0.29
0.32 0.01
0.50 + 0.06
0.68 + 0.31
0.48 0.01
GC
0.07
0.57 +
0.47 0.13
0.10
0.09
DYNOTEARS
0.02
0.00
0.63 0.01
0.52
0.58 0.03
0.00
NTS-NOTEARS
0.17 = 0.00
0.00
0.36 0.08
0.45 + 0.07
0.54 + 0.13
0.42 0.10
PCMCI+
0.63 = 0.09 0.09
0.15
0.01
0.68 + 0.02
0.64 + 0.05
0.73 0.02
Rhino
0.59 + 0.04
0.002
0.08
0.70 1 0.07
0.66 + 0.07
0.69 -: 0.04
CUTS
0.43 + 0.07
0..05
0.04
0.68 + 0.02
0.64 + 0.05
0.73 0.00
AUPRC
CUTS+
0.76 + 0.03
0.55
0.78
0.68 1 0.02
0.62 + 0.08
0.73 0.00 0.00
Neural-GO
0.66 = 0.01
0.06
0.62
0.68 + 0.01
0.55 0.08
0.64 1644000
NGM
0.75 = 0.03
0.09
0.81
0.77 0.05
0.73 0.11
0.80 + 0.05
LCCM
0.68 0.01
0.77 0.00
0.58
0.67 1 0.00
0.78 0.00
0.57 0.00
<SRU
0.76 0.01
0.77
0.70
0.78 1: 0.02
0.76 0.10
0.81 0.00
TCDF
0.002
0.35
0.65
0.37 + 0.03
ICA-LINGAM
0.337
0.30
0.33
0.61
VARLINGAM
0.06
0.33 0.35
0.39 + 0.04
0.36 + 0.06
0.44 +
0.27 0.02
LiNM
0.37 1 0.07
0.53 0.39
0.30 1: 0.01
0.37 1 0.06
0.52 1: 0.42
0.27 1: 0.01
0.0
0.1
0.2
0.3
0.4
0.5
Difference (absolute value)
Regarding the performance of the algorithms, three of them namely, PCMCI+, CUTS, and CUTS+ achieve the best tradeoff between AUROC and AUPRC on all datasets. Additionally, NGM achieves satisfactory results on the Hénon and Rivers datascts, yiclding AUPRC values among the highest. Instcad, Neural-GC performed well only on the synthetic dataset of our benchmark, while cSRU performed well only the Rivers dataset. While LiNGAM-based approaches struggled with the non-linear Hénon dataset,
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ICA-LINGAM and DirectLiNGAM demonstrated competitive performance on the real-world datasets (Rivers and AQT), particularly in terms of AUROC. Among the constraint-hased approaches, only PCMCT+ achieved satisfying performances, while, in general, the Granger- bassa approaches proved to be the best ones. None of the methods got an AUROC lower than 0.5, meaning that there were no inverted classifications. The overall performance of lested algorithms is lower than what has been reported on simpler synthetic datasets, such as Lorenz-96 (Cheng et al. 2023 Tank et al. 2021), where some methods achieved near-perfect scores. This performance gap may suggest that current algorithms are still imprecise for certain samples and datasets, and they could be further refined to improve accuracy. In general, more challenging synthetic datasets should be used to rigorously test and potentially improve existing TSCD methods. The performance degradation observed in some algorithms when exposed to new data further underscores the need for this approach.
Dealing with Latent Confounders. The existence of latent factors, called confounders, that influence both the independent variable (the cause) and the dependent variable (the effect) may lead to spurious associations, making it harder to determine the true causal relationship. Our generative model is designed faithfully replicate the underlying characteristics of the input real dataset, including any statistical and causal propertics that are present. As such, if the real dataset exhibits associations induced by latent confounders, synthetic data generated by our framework will reflect these characteristics as well.
Schur & Peters (0244) conducted an experiment in which the observed variable X is modeled as an Orn- Uhlenbeekk (OU) process, the hidden confounder U as an independent spectrally-sparse OU process, and the effect time-series Y according to the equation Y, = 3X,+ + U. + it; where 1) is i.i.d. Gaussian noise. The objective is to infer the true causal effect of X on Y, i.e. the estimation of B. The authors show that their DecoR method achieves a lower Mean Absolute Error (MAE) in estimating 3 with respect to a standard Ordinary Least Squares (OLS) approach.
Building on this framework, we assessed the ability of DiffCATS to preserve the relevant statistical and causal properties of data generated from the true OU process. Specifically, we trained the generative model on data simulated according to the above OU processes, with the parameters described by |Schur & Peters (202)). We evaluate how well causal effect estimation methods could recover 33 on both real (directly simulated) and synthetic (generative model produced) data. The results in Table 3 indicate that the generative model is able to retain the essential statistical and causal properties of the original OU-based data, as evidenced by the consistent estimation performance of both OLS and DecoR.
Table MAE in estimating B.
Real
Syuthetic
OLS DecoR
0.76 0.23
0.78 + 0.19
0.16 + 0.14
0.19 = 0.15
These findings show that: (i) existing deconfounding algorithms can be effectively applied to our synthetic data; (ii) synthetic datasets produced by our approach can serve as valuable testbeds for developing and validating new algorithms targeting latent confounding in observational data.
6.2 Causal Prediction and Classification
To demonstrate the utility of our datasct beyond benchmarking and improving causal discovery algorithms, we introduce the following two downstream tasks:
Causal Prediction. We show that conditioning a prediction model on the causal graph (see example in Figure 2 significantly improves time-series reconstruction performance, underscoring both the utility and coherence of our graphs. In detail, we trained a 2-layer LSTM predict the i-th feature of time-scries given the remaining features and the corresponding causal graph. Results are shown in Table 4 Additional details are reported in Appendix C.10
Causal Classification. Classifying observational data (i.c., time-scries) based on the underlying system dynamics (i.e., causal graphs) is a critical task across numerous domains. Indeed, causal
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classification has broad applicability, ranging from emergency detection (i.e., identify transitions to unsale system dynamics from time-series data) to customer segmentation (iee. assign customers to -eriiiiees-stie meaningful segments based on their responses to incentives - such as purchase history, engagement, or response to promotions - modeled as causal graphs). We demonstrate that our synthetic datasct can ively support (augmentt the training of classífication models, particularly scenarios where certain classos are underrepresented in real-world data. For demonstration, we organized the synthetic and real Rivers data into 11 distinct classes according to their causal graphs. We then trained 2-layer LSTM to classify unseen real time-series, achieving an Fl-score of 0.69, which highlights the utility of synthetic data. All the details are described in Appendix C.11
Table 4: Downstream task- Causal Prediction (Mean Absolute Error).
Predictor
Dataset
w/ Causal Graph
w/o Causal Graph
Hénon
0.13 +: 0.01
0.19 + 0.01
Rivers
0.01 + 0.00
0.02 = 0.00
7 Conclusions & Limitations
We introduced DiffCATS, a novel pipeline to generate faithful timo-serics along with realistic and coherent causal graphs specifically suited for the TSCD task. To the best of our knowledge, this is the first work to incorporate the simultaneous generation of causal graphs for causally related time-series generation without. the need for the stationarity assumption. We demonstrated that our model can effectively generate synthetic datasets to support the causal discovery community in enhancing their algorithms in various domains, lcarning directly from real-world observational data.
We acknowledge that one limitation of this work is that only linear causal relationships are present in the syuthetic samples. In Appendix C.9 we show a possible solution to accommodate non-linear cansal links by increasing the polynomial degree employed by the VAR. We also note that DiffCATS expressivity depends on the chosen maximum lag Tmax, which bounds the longest temporal dependencies the model can represent. Finally, the requirement of generating both the time-series and the causal graph could make its time-series-only metrics trail specialized generators (namely CSDI) and its inference slower due to iterative diffusion sampling with respect to competitors like CR-VAF. So practitioners may prefer CR-VAF for fastest sampling (but with fixed causal pattern for all synthetic samples), CSDI/timo-scries-nn when graphs are unnecessary, and DiffCATS when coherent per-sample graphs are needed without expensive post-hoc explanation.
diffusion models
In future works this approach can be extended to add a loss- based guidance of the coefficients so that the prior-known causal graph. In fact, a key advantage of our approach is
generation can be conditioned the realism and flexibility that diffusion models provide, which allows the implementation of sophisticated conditioning strategics on trained models.
Acknowledgments
This paper was prepared for informational purposes in part by the Artificial Intelligence Research group of JPMorgan Chase & Co. and its affiliates ("JP Morgan") and is not a product of the Research Department of JP Morgan. JP Morgan makes representation and warranty whatsoever and disclaims all liability, for the completeness, accuracy or reliability of the information contained herein. This document is not intended as investment research or investment advice, or a recommendation, offer or solicitation for the purchase or sale of any sccurity, financial instrument, financial product or service, or to be used in any way for evaluating the merits of participating in any transaction, and shall not constitute solicitation under any jurisdiction or any person, if such solicitation under such jurisdiction such person would be unlawful.
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A Theory
A.1 Assumptions
Our work makes the following assumptions, aligned with several TSCD algorithms:
Markovian Condition: The joint distribution of the multi-variate time-series can be factorized into P(x) II, ((x))) i.c., every variable dependent only on its parents.
Causal Faithfulness: It assumes that the relationships between variables in the data faithfully reflect the true causal connections between them.
On the other hand, thanks to the fact that our approach generates the time-series and its strictly associated causal graph, we do not need to assume causal stationarity:
Causal Stationarity: It states that all the causal links do not change over time.
A.2 Relation of Definition 1 to Granger Causality and Transfer Entropy
In this section we provide additional details and intuitions to clarify how our Definition ! relates to established notions such as Granger causality (GC) and Transfer Entropy (TE).
The intuition behind this as follows:
Connection to Granger causality. Our method parameterizes inter-variable temporal dependencies using VAR-type structure, which is the same modeling family typically used to define linear GC. When trained with an L2 (MSE) objective, the resulting estimation is closely related to least-squares fitting of a lincar autoregressive model. Under standard preprocessing/ regularity conditions (c.g., slandardized variables and limited predictor collinearity), larger VAR coelficients tend to correspond to predictors with larger marginal contributions in reducing prediction error.
2. GC-TE equivalence under Gaussianity. We rely on the result of Barnett et al. (209)) which shows that for jointly Gaussian processes (equivalently, linear models with Gaussian residuals), GC and TE quantify the same directed dependence up to constant factor.
Connection to Granger causality. Our formulation identifies candidate links by thresholding the magnitude of learned VAR coefficients. Let (t) denote the coefficient multiplying 21(1- in the predictor for (I) at lag 7 and time index (with 7 E (1,..., (l....,,mmx>) )). In our framework, these coefficients arise from the causal reconstruction module, while the overall model is trained to minimize an L2 reconstruction loss: Lnee | - f.
Minimizing MSE encourages the model to fit the conditional mean of cach target variable given its predictors, which is consistent with least-squares estimation of a linear VAR. Moreover, under a linear- Gaaarraassiinn noise interpretation (approximately Gaussian, homoscedastic residuals), minimizing squared error is equivalent to maximizing a conditional Gaussian likelihood.
In linear VAR formulations, GC is defined via the improvement in the prediction of a target when including the past of a candidate driver, relative to restricted model that excludes it. Concretely, consider predicting the target coordinate (1) from the past window (a((-1),.... ((l--)))......"lllnaa [m))]) Let the full linear predictor include all variables' histories up to Tmax, and let the restricted predictor remove the regressor corresponding to x; T) (or, in a stronger form, remove all lags of x.)). The "no directed influence at lag r" null can be expressed as Ho : dيع = 0, (and the "no Granger causality from i to up to Tiaxx" null as Ho ciu = 0 -V E {1...., })- (
In our sample-specifie, time-worying setting, the natural analogue is local in time: Ho(!): of (1) = 0. Moreover, since Definition 1 aggregates coefficient magnitudes over time using a quantile operator, an "absence of link" consistent with our extraction rule can be stated as: Ho: : ((lc) (I)) Tmax S I L}) S E, for a small g and Q.(-) denoting the q-quantile.
Var(e]"r) A common definition of linear GC strength is: Fry log Var(efu1y. where efalll and 2i denote the residuals of the optimal full restricted linear predictors (with analogous multivariate generalizations using covariance
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determinants). Our model is trained to minimize the reconstruction loss (Section 4.1), and the causal reconstruction produces & through the VAR coefficients. Minimizing squared reconstruction error is precisely the criterion that (in a linear regression interpretation) minimizes residual variance. Thus, in the linear prodictive regime, the learned coefficients (d)(()) paramcterize the same type of conditional-met mean predictor that underlies Granger-style predictabiliiy.
We recognize that, our Definition 1 is not an explicit computation of Fi-jj nor formal hypothesis test. Instead, it is practical edge-selection rule based on the magnitude and persistence of the learned coefficients. Summarizing, this is consistent with GC in the following sense:
If (1) = for Tiaa I l S L, then (2((c (1)) = 0 and the corresponding link will not be selected, matching the Granger-style "no predictive contribution" null.
Under standardized variables and weak predictor collinearity, (e)) serves as practical proxy for the predictive importance of regressor rl-(-r) in the conditional-mean model for zy((): larger (0)) typically associated with a larger increase in residual variance when that regressor removed, and hence with larger Granger-style predictability gains. We explicitly state this as an interpretability prozy.
Equivalence of Granger causality and Transfer Entropy (TE). Standard GC measures how much the prediction-error variance decreases when a variable (or its past) is added to the predictor set. Transfer entropy (TE) can be written as a conditional mutual information:
Ta = IXxpant X1(1) X
where past denotes the past of all variables except (up to Tmax).
Barnett et al. 2009 show that for jointly Gaussian processes (equivalently, linear models with Gaussian residuals), linear GC Fi-s and TE Ti-j quantify the same directed dependence up to constant factor. Using natural logarithms, this relationship can be written as Fi-j = 2T (with the constant changing if a different logarithm base is used). Therefore, within the linear-Gaussian approximation, stronger directed predictability gains (GC) correspond to stronger information flow (TE). Consistent with the discussion above, our cafffciinntthressollee rule should be interpreted as a practical proxy for such directed dependence regime, rather than an explicit computation of GC/TE in full generality.
the
B Implementation Details
B.1 Datasets
Table reports the most important statistics of our datasets and Figure 4 shows the real causal graphs of the datasets. We also include additional details not discussed in Section 5.1
Table 5: Statistics of the datasets.
Number of Training Samples
Sequence Length
Number of Variables
Number of Causal Relations
Dataset
Hénon
11295
32
11
Rivers
AQI
7246
24
36
354
Hénon: As described by Li ct al.
2023) it is generated by the following equations:
x x 1.4 - (x!) + 0.3 x
x 1.4 (e x i-1 (1 -) c) x + 0.3 x
with 2.....d. d, where the number of dimensions 6 and e 0.3. The initial values are sampled from a standard Gaussian distribution. In this dataset, the causal graph consists of positive relationship with
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X1
Dillingen
X²
Kempten
2
Lenggries
#
(a)
Hénon
(b) Rivers
Figure Real Causal Graphs (green and red arrows represent positive and negative causal relationships, respectively).
lag of 2 between a variable and itself (xi-2 + 21) and two negative relationships. The first one is between the variable and itself with lag of 1 (-x,-1 + a(); the second one is between two consecutive variables again with lag of 1 (-#f-1 + 1.
Rivers: The data are provided by the Bavarian Environmental Agency:
Air Quality Index (AQI): can be found at: https betps://vrt http://www..oooo...oooo.m
trbaa-cooputtinn We recall that, as in (Cheng et al. 2024) state, the causal relations the AQT dataset. are highly dependent on geometry distances. The graph contained in the dataset they released has been extracted considering the Gaussian kernel and threshold with respect to the geographic distances of the sensors. In particular,
{
1, o
dist(i, S
Way =
otherwise
where dist measures the distance wetween two sensors and a is set to ss 40 km. See the work of Cheng et al. 2024 for more details.
B.2
Benchmark
We compare our approach against two state-offthe-att approaches, whose code is available from the respective repositories:
TIUSALTIME (Cheng et al. 2024 It is an auloregressive model based on normalizing flows, able to observe some time-steps of the time-series and generate the subsequent step. Thanks to this architecture, the authors can extract the importance cach fcature in the input time-serics using an explainability technique,i.., DeepSHAP, provided by Sundararajan Najmi (2020). and eventually extract a causal graph.
https::/ggitie com/jarrycyx/UMN ht
CR-VAE LLi et al. 2023 https://githhn http:://gg.hh...omm///ggggggg/g/ com//hongmiing1119999//--
is based on a recurrent VAE made up of a multi-head decoder, in which the p-th head is responsible for generating the p-th feature of the time-series. Encouraged by sparsity penalty on the weights of the decodor, learns a sparse causal matrix able to encode causal relationships among the variables. Since the causal matrix is part of the model's parameter, will be the same for each synthetic sample generated by the model, in contrast. to our approach and CAUSALTIME. Moreover, a notable limitation of CR-VAE is that it is restricted to the notion of Granger Causality, implying that it does not consider the concept of time lag in observing the cansal relationships.
CSDI Tashiro et 2021
https: sttps://github /ggggttt.b..... hamm///moorrrop/(S// CSDI We modified the original CSDI model to make it suitable for the unconditional time-series generation task, following (Coletta et al.
2023
We tuned the hyper-parameters of these models on all the datasets and they are reported in Table
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Table 6: prrrattarsseeerss of CAUSALTIME, CR-VAE and CSDI.
Dataset Rivers
Model
Hyper-parameter
Hénon
AQI
Share type
Decoder
Denoder
Decoder
Epochs Train Phase
10
N. Epochs Train Phase 2
10
10
5
Learning
0.001 32
0.0001
0.0001
Batch size
32
32
CIME
Hidden slze
128
N. Layers
N. Heads
Dropout
Flow length
Hidden
61
61
64
N. Iterations Train Phase 1
1000
1000
1000
CR-VAE
N. Iterations Train Phase 2
90000
9000
90000
Learning rate
0.05
0.05
0.05
Batch size
1024
1024
1024
Epochs
Batch size
16
16
CSDI
Learning rate @ Schedule
0.0001
0.0001
0.0001
Quadratic
Quadratic le - 06
Quadratic
@ Start
le 06
1e - 06
End
Diffusion Timesteps (T)
50
50
50
B.3
Architecture Denoising Network
Here we discuss the denoising network architecture. We slightly modified the architecture of the work of [ong et al. (202)) released in the authors' repository by adding the convolution layer output the coefficients. The overall architecture of DEN, is depicted in Figure 5 It consists of an initial convolution layer and series of RESNET and ATTENTION blocks shown in Figures and respectively.
To represent the denoising time-step the model employs cosine embedding and an MLP block made up of linear layers with the activation function f(z) = x (x) in the middle, where a represents the SIGMOID lunction (xx) Ite-a. The time-step information is injected all the RESNET blocks.
In the pictures, NON-LINEARITY and GRoUPNORM refer the function f(z) and Group Normalization, rospectively. The DOWNSAMPLE block is just Id-convolution with stride equal to 2. The UPSAMPLE block is made up of Nearest Interpolation and ld-convolution.
The end part of the architecture is made up of d convolutional layers. The first one is responsible for outputting the first Tmaz steps of the timo-scrics. Each of the remaining d convolutional layers responsible for the coefficients related to the causality impact on one feature. Then, the coefficients are multiplied with the initial steps of the time-scries and the final output reconstructed following the formalization in Section 4.2
The most important hyper- pagerreaameterss reported in Table 7
Table 7: parameess of the generative model.
Training epochs
Batch size
Learning
Diffusion Timesteps (T)
Time-step
schodule
rate
embedding
Lineur start=0.0001, end=0.02
Cosine
din=129
B.4
Evaluation Metrics
In this section, describe each evaluation metric in detail.
DISCRIMINATIVE SCORE YYon et al. 2019 It measures the fidelity of synthetic timo-scries, evaluating to which extent they are indistinguishable from real ones. consists in training an off-the-shell 2-layer LSTM
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Embedding
MLP
Nx
Resnet Block
Attention Block
Resnet Block
Attention Block
Convid
Downsample
Upsample
Attention Block
Resnet Block
Attention Block
Resnet Block
Resnet Block
GroupNorm Non-linearity
Convid
Convid
Convid
Coefficients
Coefficients for feature df
feature
Shape: dd T, L
Shape: [d T
First r steps of the time-series Shape: [d,
Concatenate
Time-series Shape: dd, L]
Figure 5: Architecture of DEN
GroupNorm Non-linearity
Convid
GroupNorm Non-linearily
Drapout
Convid
Non Nin-lincariyy
Linear
Figure 6:
Architecture of the Resnet Block of DENg.
to distinguish real samples from synthetic ones. The model is trained for 30 epochs with a learning rate of Ie 4, hidden size equals 8, and batch size set to 32, The loss function to be optimized is the BINARY CROSs ENTROPY where real samples are labeled 1 and synthotic samples The score is formally defined 0055 AUROC), where AUROC the area under the ROC (Receiver-Operati Characteristic) curve of the trained discriminator.
PREDICTIVE SCORE (Yoon et al. 2019: It measures the usefulness of synthetic time-series for downstream prediction task. Following the train-on- coin-oon-synen on-sythheiic and test-on-real criterion, a post-hoc sequence- sequence-prediction model is trained to predict the subsequent steps of a time-series on synthetic data and evaluated on real data terms of the Mean Absolute Error (MAE) of the reconstructions. We trained a 2-layer LSTM-based predictor forecast the last to seq_len time-steps over each synthetic sample for 10 epochs, with a learning rate of 1e 3, hidden size equal to 32, and batch size set 32. The loss function be optimized is the -loss. Then, the predictor is evaluated Absolute Error (MAE). Formally, given real sequence æ of length seq_len let &first and Biast be the first 9 len and the last To seq_ len time-steps, respectively. The predictor observe x first and
real data, and the error is quantified using the Mcan
10
Attention Layer Figure 7: Architecture of the Attention Block of DEN,
GroupNorm
Conv1d
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predicts the subsequent 10 len time-steps, denoted as Fredd The MAE-based performance consists of len
[ (1) #preed())
seg len
AUTHENTICHTY Alaa et al. 2022 It measures the proportion of synthetic data A E [o, 1).1] that is authenlic, i.e the models should not simply memorize the training dataset by generating copies of real samples it has observed, should instcad create new, original samples. We considcred the original implementation provided by the work of Alaa et al. (022).. The metric evaluated through a hypothesis test for data copying, which employs a nearest-neighhor classifier. A synthetic sample is considered unauthentic training sample. A score close to 1 indicates that the model generating novel, unseen data.
is closest to a real
MAXIMUM MEAN DISCREPANCY (Gretton et al., 20060: It measures the similarity of synthetic and real time- series distributions. Formally, it is defined as MMD"(P, Q) = Er(k(X, x)] 2 Br. [&(X, Y)] + B Y)] where k(:,-) is the Radial Basis Function (RBF) kernel. We used the scikit-learn implementation of the RBF kernel.
COROOSSCRRRELTIONN: measures the extent to which multiple synthetic time-series preserve the cross- correlation present in real data. Specifically, we evaluate the MAE between the correlation values of the real and synthetic features. We computed the Cross- Correlation distance for each lag up to 4. Formally, let and & be a real and synthetic sample respectively. Moreover, let æ; and é; be the i-th feature of the real and the synthetic sample (A1 I i S d), respectively. The score is formally defined as Σi= (1)
E E(a.)e ((ttg))) (ax, (z)(T) (0; * (T)(, where (æ, (T) denotes the cross-correlation between æ; and Øj with respect to lag т.
C
Additional Results
C.1
Samples
Figure 5 visually shows the algorithm for extracting the causal graph from the coefficients samples of the Rivers dataset, according to Definition ! In the figure, (a) and (b) are two syuthetic samples with different causal dynamics. Figure (c) shows the coefficients generating the Dillingen feature. In particular, for both samples, each histogram considers the impact of each of the other features on Dillingen. For instance, the pink histograms represent the impact of Kempten with a lag of time-step on Dillingen since they are the coefficients of the time-step lagged Kempten that are used to generate Dillingen. Similarly to Figure (c), Figures (d) and (e) show the coefficients weighting the impact on the Kempten and the Lenggries features. The dashed black line in the figures is the q-quantile (q 0.95) in Definition ! employed to quantitatively evaluate the generated coefficients over the time-scries time-steps. All the q-quantiles are then reported in (f) and (g), for Sample A and Sample B, respectively, where the highest (in terms of magnitude) coefficients are selected to get the p% sparsity parameter used in Definition [ Indeed, the dashed black line in Figures (f) and (g) is the threshold such that only the p% highest values are selected over the entire set of generated samples.
Figures g and 10 show additional examples of real and generated samples for the Rivers and Hénon datasets, respectively.
C.2 Additional ablation studies
Table 8 show the quantitative results for the other variants of our model, highlighting the impact of each loss component. Additionally, DiffCATS w/L1 w/DTW considers a DTW-based loss and fi-norm to regularize the coefficients; DifECATS w/L.2 W/FOURIER considers Fourier-based loss (weighted by term Xg 100), and no-or to regularize the coefficients. We considered the Fourier- based term employed by Yuan & Qiao
2024):
F (D, Zo: 0) [FFT((a) - -FF (3)
(4)
where FFT() indicates the Fast Fourier Transformation EEliott & Rao 1982)
hhttps//scckkt-leamm.
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102
Kompton Lenggries
Dilingnn
203
Dillingen
300
Dillngen Kempten
250
Kemplen
Kempton
200
enggrres
152
Langagries
325
Lanagaiex
100
Timoetapa
Timestass
(a) Sample A
(b) Sample
Dillingon y-2
Dillingen
Kempten r=2
Kempton- t=1
Longories
Lengeries
Sanpure
Impact Dillingen
(d) Impact Kempten
(c) Impact on Lenggries
0.5-
0.6-
Density
0.10
0.15
0.20
0.25
0.30
0.35
0.10
0.15
0.20
0.25
0.30
0.35
Quanitles
Quantiles
(f) Sample A Quantiles
(g) Sample B Quantiles
Figure 8: Procedure to extract the causal graphs from the coefficients.
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Dillingen
Kempten
Lenggries
Real
Real
DiffCATS w/L2 w/DTW
DIffCATS w/L2 w/DTW
500
250
200
300
175
250
200
125
150
100
75
10
20
30
10
20
30
10
20
10
20
30
CausalTime
CausalTime
CR-VAE
CR-VAE
450
450
400
400
350
300
250
300
250
250
200
200
200
100
100
100
10
20
30
10
20
30
10
20
10
20
30
Figure 9: Examples from the Rivers datasel.
XO
X1
X2
X3
X4
X5
Real
Real
DifICATS w/L2 w/DTW
DifICATS w/L2 w/DTW
2.0
1.5
1.0
2.0
0.5
-0.5
0.5-
1.0
-1.0-
-2-
-3
-2.0
-2.0
20
20
30
10
20
30
10
20
30
20
20
30
CausalTime
CausalTime
CR-VAE
CR-VAE
2.0
2.0
2.5
2.0
1.5
1.5-
2.0
1.5-
1.0
2.0
1.5-
0.5
0.5-
1.0
0.0
0.0
0.0
0.0
-0.5
-1.0
-1.5
Figure 10: Examples from the Hénon dataset.
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Table 8: Results of other models on the three datasets. + indicates lower is better and 1 indicates higher is better.
Models Time-Series & Causal-Graphs
Dataset
Metric
DiFfCATS
DiffCATS w/DTW
DiffCAtS
DiffCATS w/1.2 w/Fourier
DiffCATS
w/1.2
w/L. w/DTW
Discr.
0.041 0.026
0.045 + 0.025
0.038 +: 0.022
0.035 : 0.015
0.040 = 0.024
Prod.
0.216 0.010
0.217 1: 0.008
0.216 1 0.010
0.217 1: 0.008
0.217 1 0.0010
Auth.
0.603 0.006
0.713 0.000
0.662 +: 0.007
0.677 0.008
0.700 0.006
Hénon
MMD
0.001 0.000
0.001 0.000
0.001 + 0.000
0.001 = 0.000
0.001 = 0.000
xCorr.
0.003
0.035 0.004
0.029 +:
0.037 0.005
0.035
GC-FPR.
0.403 0.001
0.421 0.001
0.423 - 0.001
0.397 = 0.000
0.403 0.001
Graph-FPR.
0.090 0.000
0.000 0.080
0.023 + 0.000
0.040 0.000
0.090 0.000
Diact 4
0.080 0.010
0.130 0.000
0.070 + 0.023
0.480 = 0.010
0.100 = 0.010
Pred.
0.036
Auth.
0.580 0.010
0.620 1: 0.010
0.585 1 0.012
0.870 1: 0.020
0.610 0.010
Rivers
MMD
0.001 0.000
0.001 0.000
0.001
0.054 0.003
0.001
xCorr.
0.000 0.000
0.060 0.010
0.022 - 0.004
0.080 0.010
0.070 0.010
GC-FPR. +
0.230 0.000
0.220 0.000
0.158 +: 0.001
0.270 1: 0.000
0.230 0.000
Greph-FPR.
0.100 0.000
0.070 0.000
0.074 + 0.002
0.160 0.000
0.070 0.000
Discr. 4
0.430 0.000
0.380
Pred.
0.048 0.001
0.048 0.000
0.060 + 0.002
0.049 0.001
0.050 0.001
Auth.
0.810 0.020
0.000
0.883 - 0.008
0.820 0.010
0,810 0.010
AQI
MMD
0.001 0.000
0.001 0.000
0.001 + 0,000
0.001 0.00G
0.001 0,000
xCorr. 4
0.090 0.010
0.110 0.010
0.100 1 0.003
0.130 1 0.010
0.100 0.010
GC-PPR.
0.480
0.390
Graph-FPR.
As the results show, incorporating the fz-norm of the coefficients into the objective loss as attempt. to sparsify the causal graph reduces the number of wrong connections. Moreover, the DTW-based loss considerably aids in extracting synchronization signals among the temporal sequences, significantly improving overall performance.
C.3 Sensitivity Analysis of Lag Parameters Tmar
In this subsoction we perform a sensitivity analysis by varying Tmaz on the rivers dataset. In general, the parameter Tmar should be selected according to the time resolution and the expected causal dynamics of the underlying phenomena. However, when the true value is uncertain, a slight overestimation can be safely adopted. As shown Table g the model maintains strong performance for both the generated time series and the causal graph, even when Tmaz is overestimated or deviates from the true lag of the causal relationships (the maximum ground-truth lag is 2). This demonstrates the model's robustness in generating reliable synthetic data, even with imperfect domain knowledge.
Table 9: Model performance under different lag values.
Tmar
Discr, score Pred. score Graph-FPR
0.099
0.032
0.022
0.067
0.033
0.070
0.060
0.031
0.078
0.050
0.033
0.086
C.4 Dimensionality Reduction
It is used to evaluate the diversity of synthetic samples, i.e., they cover the full variability of real samples. We employed t-SNE (Van der Maaten & Hinton 2008) and PCA (Bryant & Yarnold 1995) on both real and synthetic data to easily visualize how similar the two distributions are in a 2-dimensional space. We used the scikit-learn implementation for both PCA and 1-SNE. For each sample, we flattened the dimension of the features by computing the mean.
Figure 1 and Figure 1 show the t-SNE and PCA plots of our best model against the seate-of-tho-ard approaches. It. be observed that the distribution of the synthetic samples closely resembles the real one
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Base Dituston
CSDI
OIFICAT wiL2 WDTW
ClussiTlme
CRAME
Spatatt
Cripkuc
Crigial
(a) Hénon
Bose Diflusion
CSDI
DFFICAT WiLk WOTH
Cousa Tise
CRAAE
Spathet
(b) Rivers
Base Dimrutonn
CSDI
DIICATS WL2 wOTW
CausafTime
CRASE
(c) AQI
Figure 11: Dimensionality reduction: t-SNE.
Creeal
Syntheli
Systmali:
(a) Hénon
gival
(b) Rivers
Base
wDTW
CousalTlme
Ongnd
Orghel
60
Systhate
(c) AQI
Figure 12: Dimensionality reduction: PCA.
in two of the three datasots (Hénon and Rivers). This visually ensures that the model generating realistic time-series in diverse set of fields. Figure 13 and Figure 14 show the dimensionality reduction results for the other variants of the model all considered datasets.
C.5 Kernel Density Estimation
Figure 15 shows the data distributions drawn from kernel density estimation (KDE) of DiffCATS against the state-of-the-art approaches (ablations in Figure 16). KDE provides visual assessment of the model's ability to capture the marginal distributions of the real data. A key insight from these plots is the high degree of overlap between the Original real data (red) and DiffCATS' data (blue) curves. This indicates that our model faithfully reproduces the statistical properties of the underlying data, including multi-modal distributions and peak densitics. In contrast, bascline methods such as CR-VAE often exhibit smoother distributions that fail to capture the sharper density peaks observed in the Rivers and Hénon datasets, suggesting a tendency to over-regularize average out specific data characteristies. The tight alignment achieved by DiffCATS
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DIICATS
BITCATT W1L2
OFTICATS WiDTW
DITCATS MEL WIDTW
DEFICATS wiL2 wiFourirr
Orginal dhete
Original Syrtte
by thete
Synthati
Systhetc
(a) Hénon
DIRDATT
DIECATS w11?
OETCATS wiDTWW
BIFEATS WiLL MOTH
DIFICATS MI witourter
Sytteltc
Synteatc
Syrthatt
Systhetc
Spromitc
(b) Rivers
DIREATS
DREAATS
DHIECATS wibD
BIFEATS wiLL WDTI
DIFICATS MI witouder
Orana
2005
(c) AQI
Figure 13: Ablation Dimensionality reduction: L-SNE.
DFFICTS
DINCATS WIL2
OEYCATS WiDTW
DINCATS WILL WIOTIN
DITICATT WiLL wiFoufer
Original
Sp thete
Syntheis
Synthatic
Systeiic
Hénon
DITCCTTS
DEIIAAL' wil
DiTICAT' W.TW
DElICAL' will WDTW
DiTICA'' wil2 wrfourrr
Syethite
Syethdi:
Syrthetk
Syttetic
(b) Rivers
BITCATS
DHTAAL' wil
DIFCCATS wibr
DETAALS WILL WUTW
DIICAIS wid aifourer
Orignal
Synthess
Synthets
Synthie
Spmaii
(c) AQI
Figure 14: Ablation Dimensionality reduction: PCA.
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Base Demuuiion
CSDI
DIPICATS wiL2 WOTTW
CausafTime
CRAME
Syrthop
Systeet
Oat dows (a)
041
Hénon
Case Diffusion
CSDI
DIFCATS wiL3 w.DDW
Causolt ms
CE-VAE
by dydiec
cop
340
3001
(b) Rivers
Base Diffusion
CSDI
DITICA'S ML22 WDTW
CausalTime
VEVAE
Spithatc
Systhett
com
Systhett
Syntheti
Data Marue
AQI Figure 15: Kernel Density Estimation.
DIMICATS
DIMCATS WOTW
EIYTCATT
DFFCCATS w/12 w/Fourer
Orginal
Origiall
Drebat
011 No1
040
Jes Pers
(a) Hénon
DIREATS
Origiial
gina
Original
Original
Original
220
30
20
Dala
(b) Rivers
Origital
Original
Orignal
Original
2c300
30
301
301
300
(c) AQI
Figure 16: Ablation Kernel Density Estimation.
confirms that the generated time-series preserve the fundamental distributional nature of the real-world phenomena.
C.6
Evaluation Metrics during Training
Figures 17 to 19 show the evolution of the evaluation metrics during training on the Hénon, Rivers, and AQU datasets, respectively. This offers several insights into the learning stability and convergence of DiffCATS.
First, we observe rapid convergence of the MMD and Discriminative Score, which drop significantly within the first 20 epochs. This indicates that the diffusion model quickly learns to generate samples that are statistically similar to, and difficult to distinguish from, real data.
Second, the Authenticity metric, while decreasing from an initial value 1.0 (where noise is purely novel), stabilizes at a robust level (typically between 0.6 and 0.7). This confirms that while the model learns approximate the real distribution, does not simply memorize the training set; continues to generate novel samples rather than copies.
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Finally, the causal metrics (GC-FPR and Graph-FPR) show consistent downward trend or stabilization concurrent with the time-series quality metrics. This demonstrates the effectiveness of the joint optimiza- tion strategy: the model successfully refines its understanding of the causal structure (the causal graph) simultancously with the temporal dynamics, without one objective destabilizing the other.
Authenticity
Discriminative Score
Predictive Score
1.0
V
VV
0.4
0.225
W
M
0.2
0.200
0.175
20
40
20
40
Epoch
Epoch
Epoch
MMD
Cross-Correlation
0.0015
0.2
0.0014
0.1
m
20
40
20
40
Epoch
Epoch
GC-FPR
Graph-FPR
0.0175
0.48
0.46
0.0170
0.44
0
20
40
20
40
Epoch
Epoch
Figure 17: Evaluation metrics during training Hénon dataset.
Authenticity
Discriminative Score
Predictive Score
1.0
0.4
0.4
0.8
0.2
0.2
0.6
20
20
40
Epoch
Epoch
Epoch
MMD
Cross-Correlation
0.00140
0.15
M
0.10
0.00135
0.05
20
40
20
40
Epoch
Epoch
GC-FPR
Graph-FPR
0.16
0.02
0.15
0.14
0.01
20
20
40
Epoch
Epoch
Figure 18: Evaluation metrics during training Rivers dataset.
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Authenticity
Discriminative Score
Predictive Score
1.0
M
0.5
0.15
0.10
0.9
0.3
0.05
20
40
0
20
40
0
20
40
Epoch
Epoch
Epoch
MMD
Cross-Correlation
GC-FPR
0.20
0.400
0.02
0.15
0.10
0.395
0.00
20
40
20
40
0
20
40
Epoch
Epoch
Epoch
Figure 19: Evaluation metrics during training AQI dataset.
C.7 Graph Extraction Module
C.7.1 Robustness of Graph Extraction
In this section, we report an additional experiment to highlight the robustness of the causal graphs to noise
in the time-series. In particular, we trained DiffCATS in setting where Gaussian noise 2 ~~ N0, o²) has been added to the training time-serics. Table 10 reports the results the Graph-FPR for σ² = 0.005 and x² 0.01 showing the robustness of the extraction to noise the time-series.
Table 10: Graph-FPP -PR with respect to noisy time-series.
Noise variance (o²)
Dataset
0
0.005
0.01
Hénoa
0.04 1 0.00
0.02 1 0.00
0.02 1 0.00
Rivers
0.07 + 0.00
0.01 + 0.00
0.01 + 0.00
C.7.2 Statistical test
Alternative approaches can be integrated to extract the causal graph based the VAR coefficients. For example Hyvärinen et al. (2010) suggests conducting statistical test of the significance of the coefficient. We tested the Dixon's Q Test to assess if there is at least one of the generated coefficients significantly higher than the others, resulting in a notable causal link. We evaluated the extracted causal graphs using this method with a p-value of 0.05. The Graph-FPR metric 0.016 and 0.066 for the Rivers and Hénon datasets, respectively, which is very close to the score we obtained with the approach of Definition !
C.7.3 Impact of p and q
The results in Figure 20 show that the error in introducing edges remains contained when varying q E {0.80, 0.85, 0.90, 0.95) and p {0.001, 0.005, 0.01, 0.02). As the parameter p controlling the sparsity of the datasot increases, the Graph-FPR increases well, meaning that the more relationships are kept in the dataset, the more it is the probability that they are wrong. However, it can be observed that the parameter is able to alleviate this by forcing each individual sample to keep only the coefficient. of high magnitude.
We would like highlight that the parameters p and q are used extract causal graphs for the dataset from the generated coefficients. Critically, this means that a potential user of the model does not need to retrain the model to find the best p and q values.
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68
Gra0h-FP9
FPR
3.6
0.5
5.2
0.0
0.8
58
3.005
52
0.05
(a) Hénon
(b) Rivers
Figure 20: 3D surface plots showing Graph-FPR vs g and p for Hénon (a) and Rivers (b) datasets.
C.8
Execution Time
We the experiments on machine equipped with Intel Core 19-10920X CPU @ 3.50GHz, NVIDIA GeForce RTX 2060 GPU, and 8 x 32 GB DDR4 RAM.
A training phase of our model required ~1 hour for 60 epochs. We have trained 6 variants of our model on 3 different datasets, for total of 18 training hours.
In more detail, Table shows the inference time of the models isolating the generation of the time-series and the extraction of the graph. turns that even if CAUSALTIME is faster than DiffCATS in generating the time-series, the graph extraction through DeepSHAP introduces important overload making it the slowest. model.
Table 11: Inference time.
Model
Datasct
CAUSALTIME with Graph
DEFICATS
with Graph
CAUSALTIME
CR-VAE
Hénon
1481ms
8790ms
Rivers
1425ms
1492ms
235ms
4248ms
148ms
AQI
1395ms
1535ms
1349ms
205s
442ms
C.9 Increasing the Polynomial Degree
To accommodate non-lincar causal links, the polynomial degree employed in the VAR reconstruction can be increased to approximale non-linear ionshios With certain degree of approximation, any (non- linear) continuous function can be represented by a polynomial (Stone-Weiee Weierstrass Theorem), trading off approximation error of the real function with computational time, since there will be more coefficients generated by the model. As an example, we performed an experiment on the Hénon dataset with a polynomial of degree set to since this dataset comprehends a quadratic relationship. Results shown Table 12 show that the metrics to evaluate the time-series and the causal graph remain satisfactory even by raising the degree of the polynomial.
Table 12: Results on the Hénon dataset increasing the polynomial the VAR.
Polynomial Degree
Metric
2
Discr.
0.032
0.021
Pred.
0.156
0.226
Graph-PPR
0.017
0.019
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C.10 Causal Prediction Downstream Task involving Causal Graphs
In this section, show the results of downstream task involving the causal graphs. In particular, we considered the task of predicting the i-th feature of the multivariate time-series given its other dimensions and the corresponding causal graph.
Let z/4 be the (d - vavvaae time-series consisting all the features but the i-th. The predictor model consists of a 2-layer LSTM to compute the embedding of the observed timo-scries and a linear layer graaph learn the embodding of the associated graph g. The two embeddings are summed and passod through a linear layer (Lout) lo output i;. Formally, &; (ets eg) where ets LSTM /1) and Eg Igraph The model is trained to minimize the fi-loss with respect to 22, ie. the real i-th feature of
We compared the above model with the case in which only the (d (d-1) (d-1i)vvriate time-series zya is exploited to forecast E; ie.. the model does not see the causal graph. The models are evaluated terms of Mean A Absolute Error (MAE) on an independent validation set. Table 13 shows the quantitative results highlighting that the causal graphs are useful to improve the reconstruction ability of the predictor, and Figure 21 shows
some examples.
Table 13: Downstream task - Causal Prediction (Mean Absolute Error).
Predictor
Dataset
w/ Causal Graph
w/o Causal Graph
Hénon
0.13 + 0.01
0. = 0.01
Rivers
0.01 1 0.00
0.02 1 0.00
Obs Lenggries
Obs Kempten
Target Dillingen
Causal Graph
Causal Graph
400
400
350
350
350
300
300
300
250
250
200
150
100
20
20
30
20
30
10
20
30
(a) Target: Dillingen
Obs. Dillingen
Obs. Kempten
Target Lenggries
Causal Graph
w/o Causal Graph
400
300
300
250
200
200
200
100
100
10
20
20
30
10
30
10
30
(b) Target: Lenggries
Figure 21: Causal Reconstruction.
C.11 Causal Classification Downstream Task
Classifying time-scries data based on their underlying causal dynamics represents a critical challenge the observable behaviors. The broader significance of the causal classification task lies in its ability enable meaningful grouping
machine learning, especially because the causal structure of the system influences
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of observations by their causal nature. This can be applied to problems like emergency detection, where identifying unsafe system dynamics relies on understanding the causal mechanisms driving those dynamics, or customer segmentation, where groups are formed based on inferred causal responses to interventions.
The DiffCATS framework supports this challenge by generating synthetic time-series explicitly paired with their associated causal graphs, enabling models to capture and leverage causal dependencies for classification tasks. Therefore, in this downstream task, the causal graph serves a label or defining feature for the grouping of samples based on their underlying system dynamics, rather than being directly integrated into the prediction process as in the Causal Predictive task.
As a demonstration, we used the Rivers dataset. We identified the 10 most frequent causal graphs within the dalaset and Ireated them as class labels; we also added an additional class to caplure residual or less common causal dynamics. Each class represents a distinct causal graph that reflects different patterns river discharge behavior. We then trained a 2-layer LSTM network on synthotic time series generated by DiffCATS, with the goal of predicting the underlying causal class (i.e., label) from the observed time-series. When evaluated real time-series samples, the classifier achieved an Fl-score of 0.69. This result highlights the potential of using synthetic datasets to enhance classification tasks by providing explicit structural labels.
To conchude, we believe DiffCATS provides robust support for classification tasks by generating datasets with coupled time-series and causal graphs, ensuring high fidelity in both the signals and their causality. This capability particularly valuable in scenarios where certain classes are underrepresented or entirely absent. In fact, our experiments demonstrate that the synthetic data produced by DiffCATS serve effective surrogates for training accurate classifiers.
D Algorithms
We show the algorithm reconstruct the whole timo-scrics from the output of DEN, (i.c. the initial time-steps Bstart and the set of coefficients c) in Algorithm [
The sampling procedure of synthetic couple (+. 9) is described in Algorithm
Algorithm 1 Reconstruction of a from Estart and c.
Input:
Estart,
Output: &
> Estart shape [d, Tmaa]
c. [d, d. Tmans L- Tmaa]
to Estart
for all from 0 to L Tmar do
sup +- to : Tmaz :] flatten()
c - :
I - torch. w b'a sup, c)
a. + corch.cat( [āt, x. unsqueeze --1)) dim= - 1)
end for Return 20
E TSCD Algorithms Benchmark
E.1
Related work
Recent works have studied and Lested causal discovery algorithms in several scenarios and domains. Hasan et al. (2023) provide a benchmark of 5 algorithms on both a synthetic and a real dataset, evaluating them using several binary classification metrics. Lawrence et al. (2021) use their framework to generate numerical datasets and evaluate 5 causal discovery algorithms, with an in-depth performance analysis concerning their diverse assumptions and hyper-parameters selection. Cheng et al. (2024) employs the synthetic version of
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Algorithm Sampling of (å.g).
Input: Trained denoising network DEN,
Output: 0.
ør I))
for all from T to 0 do (astant; c) + DEND (xt, 1) 8o -- RECONSTRUCT (T C) Et-1 f- Br. Vā-- 1-4, 20 +
(1- 10--a-)/aa
1-
if 1 > 0 then
2t-1 f- x;-1 + 3c
1-a 1
I-6c
end if end for 9 EXTRACYGGAPHCC))
Return &
three real datasets to benchmark 13 representative state-of-tho-art causal discovery algorithms. Finally, the very recent work of Li et. al. (2024) incorporates LLMs to discover causal relationships from observational and interventional data. Their method is compared with 4 state-of-the-art baselines.
E.2 Details on the algorithms
To evaluate the different TSCD algorithms we adapt/test them to our task using their source available code, whose repositories are listed below.
GC: Granger Causality test implemented the statsmodel library.
DYNOTEARS: http:://ghhab.coo////thaee////as
NTS-NOTEARS: http://hhhbbbccccccccggmma//////
PCMCT+: http:://gghhh..coo///ooorgggggge
Rhino: http:://gthhb.ccc/maccrotttttttt
CUTS
/ CUTS+: http://////ggggtt..oo///////////
Neural-GC: http:///ggtuaoco//a/aeeee////110
NGM: http:///hhhhhhooooooc/////////or
LCCM: httpp///gtthh.cooccccaooreeeaaat
eSRU: httes///tthhh.cccm//aaaaaaaaalaa
TCDF: httpss//gattbb.cc///aaaaaatttt
The used hyper-parameters of the algorithms reported in Table 14 (they the same for all datasets).
E.3
Benchmarking Causal Discovery Algorithms
Table 15
shows the results of our benchmark on synthetic dataset where the causal graphs are extracted
globally, following the procedure in Section 4.3
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Table 14:
of the causal discovery algorithms.
Algorithm
Hyper-parameter
Value
GC
maxlag
DYNOTEARS
nax naxr
100
lage 9. threrholld
NTS-NOTEARS
0.3
Le
PCMCI+
Tinos PC
0.01
Noise Distribution
Gaussian
Rhino
init rho init_alpha
30
laput step
CUTS
u.1
0.1 -
laput step
2
CUTS+
0.1 +
Learning rate
0.05
Neural-GC
Avidge
0.01
0.002
0.02
Steps
500
NGM
Horizon
Gl__reg
0.1
LCCM
hidden_size
20
Learning rate
#1 Learning
0.005
<SRU
Batch sise Epochs
30
500
TCDF
Epochs
1000
Learning rate
0.01
Table 15: Other results of the benchmark of Causal Discovery Algorithms. Bold and underline used to highlight the best and the second best result, respectively.
AUROC Rivers
AUPRC Rivers
Method
Hénon
AQI
Hénon
AQI
GC
0.55 0.100
0.73 + 0.16
0.50 : 0.00
0.45 = 0.11
0.54 : 0.09
0.48 + 0.08
DYNOTEARS
0.45 0.111
0.52 1: 0.08
0.50
1 0.00
0.52 0.15
0.56 1 0.08
0.51 1: 0.02
NIS-NOTEARS
0.14
0.50
0.55
0.30
PCMCI+
0.84 0.08
0.82 0.08
0.68
1 0.00
0.54 1:4:00 0.09
0.64 1 0.08
0.50 1 0.03
Rhino
0.00
0.12
0.50
0.00
0.52 0.01
0.65 0.10
0.51 - 0.03
CUTS
0.81 0.10
0.86 0.09
0.68 + 0.01
0.54
0.55 0.08
0.51 + 0.02
CUTS+
0.81 0.00
0.75
0.67 1 0.01
0.53 0.07
0.58 1 0.08
0.51 1 0.02
Neural-GC
0.67 0.000
0.07
0.50 0.01
0.52 0.01
0.53 0.05
0.48 = 0.10
NGM
0.84 0.113
0.10
0.50 1 0.01
0.63 0.16
0.81
0.12
0.47 1 0.13
LGCM
0.50 0.000
0.50 0.00
0.50 1 0.00
0.51 0.00
0.78 1 0.00
0.21 1 0.00
cSRU
0.50 = 0.0
0.10
0.50 - 0.00
0.53 0.01
0.76 0.08
0.53 1 0.01
TCDF
0.50 1: 0.0
0.50
0.50 1 0.00
0.50 0.03
0.53 0.09
0.45 0.15
F Limitations and Trade-off
Even if Table reports the performance of strong generators (e.g., CSDI), intended like-for-like comparison is against methods that can generate both (i) the multivariate time series and (ii) an associated causal graph, ideally with an explicit mechanism that promotes graph graph-sample consistency. DiffCATS is designed precisely for this paired generation setting: each synthetic sample is produced together with corresponding causal graph, and the two are structurally coherent by construction. Instcad, Base Diffusion and CSDI are optimizod only for the fidelity of the synthetic time-serics and are included as reference points, to show that DiffCATS time-series quality is not far from high-performing time-series-only generators, while additionally providing
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causal graphs. This is trade-off induced by requiring causal-graph generation and consistency, rather than as across-the-board weakness.
Furthermore, regarding efficiency, Table shows DiffCATS inference around 1.4 1.5 s/sample, which is slower than CR-VAE (hundreds of ms), consistent with diffusion sampling requiring multiple denoising steps. At the same time, DiffCATS substantially faster than CAUSALTIME, whose inference dominated by post-hoe feature-importance (DeepSHAP), reaching seconds to minutes in (e.g., up to 205s on AQI).
We also evaluated the number of GLOPs required by the models to generate synthetic sample. The results for DiffCATS include all the denoising steps (100 in our case) while the results for LAUSALTIME include all the steps involved in their autoregressive generation.
Table 16: GFLOPs per sample.
DiffCATS
CR-VAE
CAUSALTIME
Hénon
14.885
0.004
6.613
Rivers
14.819
0.002
3.301
AirQuality
15.725
0.022
22.695
CR-VAE is consistently the cheapest (0.002 0.022 GFLOPs/sample) because it produces a sample in essentially single lightweight forward pass of variational autoencoder. DiffCATS requires 14.8-15.7 CFLOPs/sample across all three dalasets due to its iterative diffusion sampling procedure. CAUSALTIME'S generation cost is competitive increases markedly on the high-dimensional AirQuality setting (22.695 GFLOPs), consistent with the fact that its autoregressive generation must be repeated across the sequence and its per-step computation grows with dimensionality.
small/moderate settings (3.301 GFT.Pss on Rivers; 6.613 on Hénon), but
Importantly, the CAUSAL'IMME GFLOPs reported here only count the forward computation needed output syuthetic sample and do not include its post-processing for extracting the causal graph by interpreting the model with APP In practice, most of CAUSAL.TIME'S wall-clock inference overhead comes from this DeepSHAP-based attribution step, which requires many additional evaluations and can dominate runtime even when the raw GFLOPs/ Sssmpee for generation looks moderate. By contrast, in our pipeline, even if DiffCATS requires more GFLOPs to generate a sample, the subsequent causal-grapl extraction step is lightweight, so the end-to-end overhead not driven by an expensive interpretability pass.
Finally, the GFLOPs trends across datasets also highlight scaling with the number of features (Rivers: 3, Hénon: 6, AirQuality: 36). DiffCATS stays roughly constant (~ 15 GFLOPs) across these experiments, while CR-VAE increases modestly (from 0.002 to 0.022 GFLOPs) and CAUSALTIME grows substantially, especially at 36 features (22.695 GFLOPs). This suggests DiffCATS generation compute is comparatively less sensitive to feature dimensionality in our implementation, whereas CAUSAL. autoregressive generator becomes significantly more expensive as dimensionality increases.
35


Reviews:
[{"review_id": "8oBjOhTvrn", "reviewer": "Reviewer_eBEv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces DiffCATS, a novel generative framework based on diffusion models. DiffCATS simultaneously generates a multivariate time-series and its corresponding ground-truth causal graph, requiring only observational data for training. The core mechanism uses a Vector Autoregressive structure within the diffusion process, where the causal graph is directly and transparently derived from the learned VAR coefficients. The utility of the generated data is demonstrated in benchmarking TSCD algorithms, causal prediction, and causal classification.\n\nStrengths: 1. The model ensures structural consistency between the generated time-series and its causal graph, which is a key objective for realistic synthetic data generation. 2. By embedding the causal structure directly through VAR coefficients, the resulting graph is naturally interpretable.\n\nWeaknesses: 1. The definition for extracting the causal graph relies on two critical hyperparameters, $\\rho$ (percentage of relationships) and $q$ (quantile threshold). I'm wondering how the two parameters affect the final performance. 2. The comparison in Table 1 shows that the proposed method still lags behind previous models in a couple of cases. Can the authors further discuss it, including the capability boundary of DiffCATS? I also notice that the inference time of DiffCATS is also somewhat long. More discussion about the balance of performance and efficiency is also needed here. 3. I find many references are not corrected cited in the paper. For example, in the first paragraph, all the citations should be in the format like (Hasan et al., 2023) rather than the current Hasan et al. (2023) because they are not a part of the sentence structure. Most citations in this paper have this problem.", "claims_and_evidence_yn": "Yes", "claims_explanation": "1. The core claims of superior realism and causal graph fidelity are backed by comparisons against multiple state-of-the-art baselines on three datasets. The paper reports superior or comparable scores on standard metrics like MMD and Discriminative Score, validating the realism of the generated data. The achievement of the best GC-FPR and Graph-FPR on all three datasets is a convincing piece of evidence that the generated graphs minimize false causal links.\n\n2. The synthetic data served as a viable testbed for benchmarking several widely used TSCD algorithms, which also supports their claim of practical utility.", "audience_yn": "Yes", "audience_explanation": "1. The paper directly addresses a critical and widely recognized bottleneck in Time-Series Causal Discovery research. The researchers in this field will find the proposed method helpful in their study.\n\n2. The paper presents a novel application of diffusion models to the multivariate time-series domain, which is of clear interest to generative model researchers.", "requested_changes": "1. Provide a sensitivity analysis and discussion on the impact of the graph extraction hyperparameters $\\rho$ and $q$.\n\n2. Supplement more discussons about some scenarios that the proposed method does not perform as well as the previous models, as well as a balance between performance and efficiency.\n\n3. For generative models, it will be very helpful if the authors can include the model size or number of flops for DiffCATS and baselines in the comparison tables.\n\n4. Correct the wrong citation formats.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "8beP47GF5g", "reviewer": "Reviewer_nzWU", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present a method that is based on diffusion models to generate time series data, alongside a corresponding causal graph related to each generated sequence. The method is trained on real data, without relying on in the specification of a ground-truth graph, but rather infers a graph that best allows for the causal modelling of the seen real data.\n\nThe authors then go on to provide experimental evidence that their generated synthetic data has merit for a number of different tasks. First, they consider three time series datasets, of which the generate synthetic samples and calculate a variety of metrics for time series generation. The proposed method performs favourably or on par with competing approaches across the board. As a second task, the authors compare a large number of causal discovery algorithms on real data and synthetic data generated by their method. In the majority of settings, models achieve a similar performance on the synthetic data as they do on real data. The authors argue that this indicates the usefulness of the synthetic data their model generates for causal discovery. As a final task, the authors show the benefit of including the causal graph generated by their method in a causal prediction and causal classification task, respectively.", "claims_and_evidence_yn": "Yes", "claims_explanation": "While the motivation for some of the experiments and claims is not crystal clear to me (see my additional comments below), the authors provide ample evidence for the claims they make in terms of experiments across many settings.\n\nTo the best of my judgement, there are no technical errors in this paper.", "audience_yn": "Yes", "audience_explanation": "The authors reference prior works which they extend, therefore I would consider the findings in this paper to be interesting to the audience interested in these works, as well.", "requested_changes": "Please tone down the claim in the first sentence of the introduction (as well as in the abstract) that real-world phenomena have an *inherent* causal structure. The description of a relationships between variables is a modelling choice and should be described as such. Whether or not the world truly functions according to causal models is an entirely different question.\n\nIf you are using the biblatex package, please use the ``\\citep`` command for citations, as the current style produces inline citations that are read as part of the text, rather than citations with clear parentheses to distinguishing them.\n\nThe caption for Figure 3 would benefit from being more verbose. Currently, I cannot understand the figure of the model from only reading the caption.", "broader_impact_concerns": "None", "additional_comments": "I cannot fully follow the motivation for generating synthetic time series for causal discovery in the way that the author's proposed method does. The concern does not seem to be overcoming artefacts of purely synthetic data such as sortability phenomena, as this is not mentioned in the manuscript. What is the benefit of generating this synthetic data? As the authors state, the goal is not to implicitly do causal discovery by means of generating data. Is the underlying motivation perhaps to have more data dor the development of causal discovery algorithms? If so, what is the benefit? Does the real data that is used to train the proposed method not suffice for this?\n\nIn my view, the criteria for acceptance to TMLR are met, but I would still be very interested in the authors' input on this topic of underlying motivation", "extra_scores": {}, "extra_text": {}}, {"review_id": "BmlUtAkYyw", "reviewer": "Reviewer_FBxn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes DiffCATS, a diffusion model–based mechanism for generating time-series data that exhibit underlying causal structures. The method is designed to serve as a benchmark for evaluating time-series causal discovery (TSCD) algorithms. The proposed DiffCATS requires only multivariate time-series data for training, even when their causal structures are unknown, and it is capable of generating synthetic time-series samples associated with diverse causal graphs. Using datasets produced by several similar generative mechanisms, the authors apply multiple causal discovery algorithms to infer causal structures and evaluate both the inferred causal graphs and the generated time-series themselves through a variety of quantitative metrics, thereby demonstrating the usefulness of DiffCATS.", "claims_and_evidence_yn": "No", "claims_explanation": "At least in its current form, the claims of this paper are not sufficiently supported for the following reasons:\n\n1. The necessity of employing a diffusion model as the generative mechanism is not adequately discussed. The authors should clarify why other types of generative models are insufficient and emphasize what specific advantages or capabilities become possible *because* of using a diffusion model.\n2. The definition of a causal relationship given in Definition 1 is crucial. In the final paragraph of Section 4, the paper asserts in words that this definition is consistent with existing formulations; however, this is a claim that warrants a formal, mathematical explanation, not merely a verbal statement.\n3. As methods for causal discovery, LiNGAM and its variants occupy an important position in the field. It is unclear why these methods are neither included in the numerical comparisons nor cited in the related work section. The authors should justify this omission.\n\nhttps://jmlr.org/papers/v7/shimizu06a.html\nhttps://sites.google.com/view/sshimizu06/lingam", "audience_yn": "Yes", "audience_explanation": "There is a consistently strong demand for causal discovery in complex real-world datasets, and thus, the development of benchmark data generation models to support such research is an essential and necessary contribution in this field.", "requested_changes": "In addition to the three issues mentioned above, I would also request the following revision:\n- In Appendices C.5 and C.6, please do more than simply present the results in graphical form. Include an explanation of what the graphs imply and what key observations or insights can be drawn from them.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)