Original: Time Series Causal Link Estimation under Hidden Confounding using Knockoff Interventions
Revised: Time Series Causal Link Estimation under Hidden Confounding using Knockoff Interventions

Original: Anonymous Author(s)

Original: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: Latent variables often mask cause-effect relationships in observational data whichprovokes spurious links that may be misinterpreted as causal.
Revised: Latent variables often mask cause-effect relationships in observational data which provokes spurious links that may be misinterpreted as causal.

Original: This problem sparksgreat interest in the fields such as climate science and economics.
Revised: This problem sparks great interest in the fields such as climate science and economics.

Original: We propose toestimate confounded causal links of time series using Sequential Causal Effect Variational Autoencoder (SCEVAE) while applying knockoff interventions.
Revised: We propose to estimate confounded causal links of time series using Sequential Causal Effect Variational Autoencoder (SCEVAE) while applying Knockoff interventions.

Revised: Knockoff variables have the same distribution as the originals and preserve the correlation to other variables.

Revised: This allows for counterfactuals that are more faithful to the observational distribution.

Original: Weshow the advantage of knockoff interventions by applying SCEVAE to syntheticdatasets with both linear and nonlinear causal links.
Revised: We show the advantage of Knockoff interventions by applying SCEVAE to synthetic datasets with both linear and nonlinear causal links.

Original: Moreover, we apply SCEVAEwith knockoffs to real aerosol-cloud-climate observational time series data.
Revised: Moreover, we apply SCEVAE with Knockoffs to real aerosol-cloud-climate observational time series data.

Original: Wecompare our results on synthetic data to those of a time series deconfoundingmethod both with and without estimated confounders.
Revised: We compare our results on synthetic data to those of a time series deconfounding method both with and without estimated confounders.

Original: We show that our methodoutperforms this benchmark by comparing both methods to the ground truth.
Revised: We show that our method outperforms this benchmark by comparing both methods to the ground truth.

Original: Forthe real data analysis, we rely on expert knowledge of causal links and demonstratehow using suitable proxy variables improves the causal link estimation in thepresence of hidden confounders.
Revised: For the real data analysis, we rely on expert knowledge of causal links and demonstrate how using suitable proxy variables improves the causal link estimation in the presence of hidden confounders.

Original:  Introduction
Revised:  Introduction

Original: Causal link estimation and inference for dynamical systems are important tasks in many fieldssuch as finance and climate science (1).
Revised: Causal link estimation and inference for dynamical systems are important tasks in many fields such as finance and climate science (1), (2).

Original: We propose to estimate both linear and nonlinear causaleffects of time seriesvariables under hidden confounding using Sequential Causal Effect Variational Autoencoder (SCEVAE) (2) while performing knockoff interventions.
Revised: We propose a novel method for estimating both linear and nonlinear causal effects in time series under hidden confounding while performing Knockoff (3) interventions, inspired by the Causal Effect Variational Autoencoder (CEVAE) (4).

Original: SCEVAE is based on a Causal
Revised: The proposed approach, as will be explained further on, allows for less biased causality analysis of non-stationary sequential data and counterfactuals that are more faithful to the distribution of observational data.

Original: Effect Variational Autoencoder (CEVAE) by Louizos et al.
Revised: A hidden confounder is a variable that influences both the cause and the effect, and its presence can lead to false relationships which may be misconceived as causal.

Original: (3), a deep learning framework whichwas first applied to analyzing effects of a binary treatment on the patients’ health outcomes.
Revised: CEVAE is a deep learning framework which was first applied to analyzing effects of a binary treatment on the patients’ health outcomes.

Original: Thecausal Directed Acyclic Graph (DAG) representing hidden confounding with one proxy variable, asused by CEVAE framework, is depicted in Fig.
Revised: The causal Directed Acyclic Graph (DAG) representing hidden confounding with one proxy variable, as used by CEVAE framework, is depicted in Fig.

Original: Y denotes the effect, W the cause, Z denotesthe hidden confounder and X a proxy variable providing noisy views on Z .
Revised: Y denotes the effect, W the cause, Z denotes the hidden confounder and X a proxy variable providing noisy views on Z .

Original: We note that the proxycan be multivariate and both categorical and continuous.
Revised: We note that the proxy can be multivariate, and both categorical and continuous.

Original: Although the DAG in Fig.
Revised: Although the DAG in Fig.

Original: 1 might appearrestrictive, in many real-world applications, such as climate science for instance, there are unobservedconfounders present, with multiple different proxy variables to describe them (4).
Revised: 1 might appear restrictive, in many real-world applications, such as climate science for instance, there are unobserved confounders present, with multiple different proxy variables to describe them (5).

Original: We assume that the causal link between the confounded variables is approximately directly identifiablethrough the proxy to the hidden confounder.
Revised: We assume that the causal link between the confounded variables is approximately directly identifiable through the proxy to the hidden confounder.

Original: We provide more details onthis and the identifiabilitytheory in Section 3.2.
Revised: We provide more details on the identifiability theory in Section 3.2.

Original: Knockoff filters were first introduced by Barber and Candès in 2015 (5) for controlling the variableselection procedure and false discovery rate of potential drivers of the response variable.
Revised: Knockoff filters were first introduced by Barber and Candès in 2015 (3) for controlling the variable selection procedure and false discovery rate of potential drivers of the response variable.

Original: Theirmethod produces knockoff variables which imitate the correlation structure of the existing variablesand do not require any new data.
Revised: Their method produces Knockoff variables which imitate the correlation structure of the existing variables and do not require any new data.

Original: At first they were limited to a linear Gaussian response variable, butthis was later extended for arbitrary and unobserved data distributions in the work by Romano et al.(6), as they introduced a framework called Deep Knockoffs based on deep generative models
Revised: At first they were limited to a linear Gaussian response variable, but this was later extended for arbitrary and unobserved data distributions in the work by Romano et al. (6), as they introduced a framework called Deep Knockoffs based on deep generative models.

Original: In the work by (2), the authors used Gaussian noise as an intervention since it removes the link tothe hidden confounder as required by do -calculus (7), but it does not preserve the distribution of theoriginal cause variable W . We therefore propose to use a knockoff (5) of W as the intervention.
Revised: Due to its highly practicable properties for counterfactual analysis (7), as proposed by (8), we generate a Knockoff (3) of the cause variable W as a way of intervention to estimate the causal link intensity using our approach.

Original: Inaddition to preserving the distribution of W , this also preserves the inter-variable correlations to otherobserved variables, but is not correlated to W .
Revised: In addition to preserving the distribution of W , this also preserves the inter-variable correlations to other observed variables, but is not correlated to W .

Original: One limitation of our approach is that we assume that,within the processed window, data is stationary and sufficient to produce good knockoffs.
Revised: To highlight the advantage of Knockoff intervention, we compare it to using Gaussian noise as intervention variable (cid:99) W . It is independent of the hidden confounder Z and therefore removes the link between W and Z as required by do -calculus (7). However, Gaussian noise intervention does not preserve the distribution of the original cause variable W , which is learned during the training of the neural networks within our method and expected for inference. This leads to suboptimal performance in contrast to Knockoff interventions.

Original: Apart from the nonlinear causal links, the problem of causal link estimation under hidden confoundingis particularly challenging for time series because the data is non-stationary, and the causal linkbetween confounded variables might be changing over time, thus introducing delayed causal effects.
Revised: Apart from the nonlinear causal links, the problem of causal link estimation under hidden confounding is particularly challenging for time series because the data is non-stationary, and the causal link between confounded variables might be changing over time, thus introducing delayed causal effects.

Revised: To address these issues, we engineered our approach to transcend the CEVAE methodology and become applicable to complicated non-stationary sequential data with the help of Long Short-Term

Revised: Memory (LSTM) (9) recurrent neural networks.

Revised: Moreover, we allow for the causal effect estimation to be done on a single time series example of each observed variable.

Revised: This means that one does not need to have multiple realizations of the variables on their disposal which is an advantage in many fields such as finance and climate science where each variable is observed only once per time step.

Original: The paper is structured as follows.
Revised: The paper is structured as follows.

Original: In Section 2, we outline related work on treatment effect estimationand provide an overview of causality analysis under hidden confounding.
Revised: In Section 2, we outline related work on treatment effect estimation and provide an overview of causality analysis under hidden confounding.

Original: Moreover, we discuss othercausality analysis work that utilizes knockoffs.
Revised: Moreover, we discuss other associated work that utilizes Knockoffs.

Original: In Section 3, we introduce the CEVAE frameworkand provide definition and discussion of causal identifiability, the assumptions under which using SCEVAE is justified, as well as a detailed description of SCEVAE methodology with knockoffinterventions and introduce knockoffs themselves in more detail.
Revised: In Section 3, we introduce the CEVAE framework, define and discuss causal identifiability, the assumptions under which using SCEVAE is justified, as well as a detailed description of our method, and introduce Knockoffs themselves in more detail.

Original: Section 4 describes the data we use,and 5 provides a detailed experimental setup along with our method’s results.
Revised: Section 4 describes the data we use, and Section 5 provides a detailed experimental setup along with our method’s results.

Original: Section 6 summarizesour approach and concludes the paper.
Revised: Section 6 summarizes our approach and concludes the paper.

Original: To the best of our knowledge, we are the first to propose  causal effect estimation  under hiddenconfounding for sequential data using SCEVAE with knockoff interventions.
Revised: To the best of our knowledge, we are the first to propose a causal effect estimation method under hidden confounding for sequential data based on CEVAE and utilize it with Knockoff interventions.

Original:  Related Work
Revised:  Related Work

Original: Treatment effect estimation in the static setting is a very active field of research (8), (9), (10).
Revised: Treatment effect estimation in the static setting is a very active field of research (10), (11), (12).

Original: Underthe assumption of no hidden confounding, methods such as VCNet (11) and its extensions by state-ofthe-art transformer networks (12) were recently applied to the task of continuous treatment estimationby Zhang et al.
Revised: Under the assumption of no hidden confounding, methods such as VCNet (13) and its extensions by state-of-the-art transformer networks (14) were recently applied to the task of continuous treatment estimation by Zhang et al.

Original: (13) and Melnychuk et al.
Revised: (15) and Melnychuk et al.

Original: In a more realistic scenario when there are unobserved confounders present, Causal Effect Variational
Revised: In a more realistic scenario when there are unobserved confounders present, Causal Effect Variational

Original: Autoencoder (CEVAE) (3) was one of the first such deep learning methods.
Revised: Autoencoder (CEVAE) (4) was one of the first such deep learning methods.

Original: It relies on the existenceof one or more proxies to directly estimate the latent confounder.
Revised: It relies on the existence of one or more proxies to directly estimate the latent confounder.

Original: Nevertheless, in contrast to abovementioned transformer-based treatment estimation methods, CEVAE focuses on binary treatment.
Revised: Nevertheless, in contrast to abovementioned transformer-based treatment estimation methods, CEVAE focuses on binary treatment.

Original: Research endeavors such as those by Rissanen and Marttinen (15), Im et al.
Revised: Research endeavors such as those by Rissanen and Marttinen (17), Im et al.

Original: (16), and Trifunovet al.
Revised: (18), and Trifunov et al.

Original: (17) either extend CEVAE to a uniform or continuous treatment, or analyse this deep latentvariable model’s capabilities.
Revised: (19) either extend CEVAE to a uniform or continuous treatment, or analyse this deep latent variable model’s capabilities.

Original: However, Sequential CEVAE (SCEVAE) (2) is the first method toextend CEVAE  to time series.
Revised: However, until now, to the best of our knowledge, CEVAE has never been applied to time series.

Original: Taking a different approach towards treatment effect estimation, Bica et al.
Revised: Taking a different approach towards treatment effect estimation, Bica et al.

Original: (18) and Hatt and Feuerriegel (19) propose deconfounding methods for sequential data using neural networks.
Revised: (20) and Hatt and Feuerriegel (21) propose deconfounding methods for sequential data using neural networks.

Original: Specifically,in the work by Bica et al.
Revised: Specifically, in the work by Bica et al.

Original: (18), the authors develop Time Series Deconfounder (TSdeconf), a methodbuilt upon RNNs with multitask output to produce a factor model over time and estimate latentvariables.
Revised: (20), the authors develop Time Series Deconfounder (TSdeconf), a method built upon RNNs with multitask output to produce a factor model over time and estimate latent variables.

Original: These latent variable estimates are then used for causal inference as proxies.
Revised: These latent variable estimates are then used for causal inference as proxies.

Original: The limitationof this approach, in contrast to SCEVAE, is that it requires many patients i.e. samples and is notsuitable for processing long time series (e.g. N ≥ 1000 ).
Revised: The limitation of this approach, in contrast to ours, is that it requires many patients i.e. samples and is not suitable for processing long time series (e.g. N ≥ 1000 ).

Original: Another shortcoming of TSdeconf is that it cannot be applied to observational data with one or more treatments assigned at each time step.
Revised: Another shortcoming of TSdeconf is that it cannot be applied to observational data with one or more treatments assigned at each time step.

Original: This was addressed by Hatt et al.
Revised: This was addressed by Hatt et al.

Original: (19), as the authors introduced a similar time series deconfoundingmethod called Sequential Deconfounder, this time based on a Gaussian process latent variable model.
Revised: (21), as the authors introduced a similar time series deconfounding method called Sequential Deconfounder, this time based on a Gaussian process latent variable model.

Original: These methods are useful when there are no proxies available.
Revised: These methods are useful when there are no proxies available.

Original: In contrast to SCEVAE with knockoffintervention, this may introduce more approximation error and requires multiple realizations of eachvariable i.e. many independent samples for training.
Revised: In contrast to SCEVAE, this may introduce more approximation error and requires multiple realizations of each variable i.e. many independent samples for training.

Original: Although SCEVAE relies on access to proxies,it is not a limitation in many fields such as environmental and climate science, where many observedvariables can be used to describe a latent one (4).
Revised: Although SCEVAE relies on access to proxies, it is not a limitation in many fields such as environmental and climate science, where many observed variables can be used to describe a latent one (5).

Original: In the recent years, due to their distribution- and correlation-preserving properties, knockoffs (5), (6)were innovatively applied for counterfactual analysis of images (20), and time series (21), (22). In
Revised: In the recent years, due to their distribution- and correlation-preserving properties, Knockoffs (3), (6) were innovatively applied for counterfactual analysis of images (22), and time series (23), (8).

Original: Ahmad et al.
Revised: Ahmad et al.

Original: (21), the authors propose to use deep learning and counterfactual knockoff variablesto aid causal discovery of multivariate nonlinear time series with lower false discovery rate thanstate-of-the-art methods.
Revised: (23) propose to use deep learning and counterfactual Knockoff variables to aid causal discovery of multivariate nonlinear time series with lower false discovery rate than state-of-the-art methods.

Original: Similarly to SCEVAE, work by Yin and Barucca (23) relies on RNNs to estimate the causal link ofthe variables under influence of a hidden confounder.
Revised: Similarly to SCEVAE, work by Yin and Barucca (24) relies on RNNs to estimate the causal link of the variables under influence of a hidden confounder.

Original: However, it does not employ do-calculus (7),but rather Granger causality (24) to determine the presence of a non-spurious causal link.
Revised: However, it does not employ do-calculus (7), but rather Granger causality (25) to determine the presence of a non-spurious causal link.

Original: Due to thischoice of causality analysis tools, it cannot detect instantaneous causal links.
Revised: Due to this choice of causality analysis tools, it cannot detect instantaneous causal links.

Original: Moreover, since we useknockoff interventions the cause variable’s distribution does not change, and its correlation to otherobservation variables is preserved.
Revised: Moreover, since we use knockoff interventions the cause variable’s distribution does not change, and its correlation to other observation variables is preserved.

Original: This type of intervention allows for more accurate causal linkestimates in comparison to standard normal interventions originally proposed for SCEVAE approachas well as lower counterfactual prediction error aswe will discuss in more detail in Section 5.1.
Revised: This type of intervention allows for more accurate causal link estimates in comparison to standard normal interventions, as well as lower counterfactual prediction error as will be discussed in more detail in Section 5.1.

Original:  Methodology3.1 Causal effect variational autoencoder
Revised:  Methodology 3.1 Causal effect variational autoencoder

Original: To better understand SCEVAE architecture, we first introduce Causal effect variational autoencoder(CEVAE) (3).
Revised: To better understand SCEVAE architecture, we first introduce Causal effect variational autoencoder (CEVAE) (4).

Original: It is a deep learning method based on a VAE (25) and a TARnet (26).
Revised: It is a deep learning method based on a VAE (26) and a TARnet (27).

Original: Its underlyingprobabilistic graphical model is shown in Fig.
Revised: Its underlying probabilistic graphical model is shown in Fig.

Original: CEVAE methodology assumes all variablesto be non-sequential.
Revised: CEVAE methodology assumes all variables to be non-sequential.

Original: W denotes binary treatment, Y an outcome of this treatment, while thelatent confounder Z , and its proxy X denote the socio-economic status and income of each patient,respectively.
Revised: W denotes binary treatment, Y an outcome of this treatment, while the latent confounder Z , and its proxy X denote the socio-economic status and income of each patient, respectively.

Original: The central aim of treatment effect estimation is recovering the Individual Treatment
Revised: The central aim of treatment effect estimation is recovering the Individual Treatment

Original: Effect (ITE) and the Average Treatment Effect (ATE) defined in (1) and (2), respectively:
Revised: Effect (ITE) and the Average Treatment Effect (ATE) defined in (1) and (2), respectively:

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: These metrics are defined for each value x of variable X , and by w 1 we denote applied treatment,while values of W when no treatment is applied are denoted by w 0 .
Revised: These metrics are defined for each value x of variable X , and by w 1 we denote applied treatment, while values of W when no treatment is applied are denoted by w 0 .

Original: In the CEVAE framework,w 1 = 1 , and w 0 = 0 .
Revised: In the CEVAE framework, w 1 = 1 , and w 0 = 0 .

Original: ATE is easily calculated once we obtain the ITE, and for that we need torecover the joint distribution p ( Z, X, W, Y ) , as stated in Theorem 1 by Louizos et al.
Revised: ATE is easily calculated once we obtain the ITE, and for that we need to recover the joint distribution p ( Z, X, W, Y ) , as stated in Theorem 1 by Louizos et al.

Original: Theorem 1.
Revised: Theorem 1.

Original: If CEVAE recovers p ( Z, X, W, Y ) , then we can recover the ITE under the causal modelin Fig.
Revised: If CEVAE recovers p ( Z, X, W, Y ) , then we can recover the ITE under the causal model in Fig.

Original: Distribution p ( Z, X, W, Y ) is obtained via CEVAE’s model network by approximating the trueposterior over Z conditioned on X , W and Y , whereas the prior p ( Z ) is modeled by the standardnormal distribution.
Revised: Distribution p ( Z, X, W, Y ) is obtained via CEVAE’s model network by approximating the true posterior over Z conditioned on X , W and Y , whereas the prior p ( Z ) is modeled by the standard normal distribution.

Original: All estimated probability distributions are parameterized by MLPs.
Revised: All estimated probability distributions are parameterized by MLPs.

Original: TARnet isused to infer the estimate of the posterior by branching for each of the two treatment groups in W .
Revised: TARnet is used to infer the estimate of the posterior by branching for each of the two treatment groups in W .

Original: Further, once all necessary distributions have been modeled, one can then construct a single objectivefor the inference and model networks, i.e. the variational lower bound :

Original: L =

Original: N (cid:88)

Original: [Equation]

Original: of the causal graphical model from Fig.

Original: By x i we denote an input data point, by w i the treatmentassignment, by y

Original: i the outcome of the specific treatment, by z

Original: i the latent confounder, and by q ( · ) wedenote estimation of the probability distribution with the same arguments.

Original: Finally, since it is necessary to know the intervention assignment W together with its outcome Ybefore inferring the posterior distribution over Z , two auxiliary distributions are introduced, helpingto predict w i and y i for new samples, so the variational lower bound becomes

Original: [Equation]

Original: where x ∗ i , w ∗ i , y ∗ i are the observed values for the input, intervention, and outcome variables in thetraining set.

Original: 3.2 Causal identifiability
Revised: 3.2 Causal identifiability

Original: When estimating causal link intensity or performing causal discovery from observational data, oneneeds to establish if the underlying model is identifiable.
Revised: When estimating causal link intensity or performing causal discovery from observational data, one needs to establish if the underlying model is identifiable.

Original: If that is not the case, a set of assumptionsunder which the identifiability holds must be imposed.
Revised: If that is not the case, a set of assumptions under which the identifiability holds must be imposed.

Original: We introduce identifiability following recentwork by Khemakhem et al.
Revised: We introduce identifiability following recent work by Khemakhem et al.

Original: and discuss this issue for deep latent variable models such as VAE.
Revised: and discuss this issue for deep latent variable models such as VAE.

Original: Definition 1.
Revised: Definition 1.

Original: Let ∼ be an equivalence relation on the set of model parameters Θ .
Revised: Let ∼ be an equivalence relation on the set of model parameters Θ .

Original: We say that a deeplatent variable model p θ ( Y, Z ) =
Revised: We say that a deep latent variable model p θ ( Y, Z ) =

Original: p θ ( Y | Z ) p θ ( Z ) , for observed variable Y ∈ R d and latent randomvector Z ∈ R n is identifiable up to ∼ if
Revised: p θ ( Y | Z ) p θ ( Z ) , for observed variable Y ∈ R d and latent random vector Z ∈ R n is identifiable up to ∼ if

Original: [Equation]
Revised: [Equation]

Original: ) for θ, ˆ θ ∈ Θ .
Revised: for θ, ˆ θ ∈ Θ .

Original: The elements of the quotient space Θ / ∼ are called the identifiability classes .
Revised: The elements of the quotient space Θ / ∼ are called the identifiability classes .

Original: The graphical model in Fig.
Revised: The graphical model in Fig.

Original: 1 is in general not identifiable.
Revised: 1 is in general not identifiable.

Original: A starting point would be to obtain modelparameters or estimate latent variables Z ∗ up to trivial transformations T i for i ∈ { 1 , . . .
Revised: A starting point would be to obtain model parameters or estimate latent variables Z ∗ up to trivial transformations T i for i ∈ { 1 , . . .

Original: , N } in theform of sufficient statistics, and invertible matrix A (27).
Revised: , N } in the form of sufficient statistics, and invertible matrix A (28).

Original: To do that, we will first introduce sufficient statistics T i with respect to the deep latent variable modelfrom Definition 1 and illustrate the procedure on a one-dimensional latent variable case.
Revised: To do that, we will first introduce sufficient statistics T i with respect to the deep latent variable model from Definition 1 and illustrate the procedure on a one-dimensional latent variable case.

Original: A statistic is sufficient for a family of probability distributions if the sample from which it is obtainedprovides no additional information than the statistic, as to which of those probability distributions thedata was sampled from (28).
Revised: A statistic is sufficient for a family of probability distributions if the sample from which it is obtained provides no additional information than the statistic, as to which of those probability distributions the data was sampled from (29).

Original: Let T i =
Revised: Let T i =

Original: ( T i, 1 , . . .
Revised: ( T i, 1 , . . .

Original: , T i,k ) , i ∈ { 1 , . . .
Revised: , T i,k ) , i ∈ { 1 , . . .

Original: , N } , k ∈ N be sufficient statistics with respect to thedeep latent variable model associated with a standard normal family of parameters
Revised: , N } , k ∈ N be sufficient statistics with respect to the deep latent variable model associated with a standard normal family of parameters

Original: λ i ( X ) =(
Revised: λ i ( X ) =

Original:  λ i, 1 ( X ) , . . .
Revised: ( λ i, 1 ( X ) , . . .

Original: , λ i,k ( X )) , given a conditioning variable X .
Revised: , λ i,k ( X )) , given a conditioning variable X .

Original: Importantly, the conditioning variable X isa proxy of the hidden confounder in our causal setup and functions λ i are parameterized by LSTMs.
Revised: Importantly, the conditioning variable X is a proxy of the hidden confounder in our causal setup and functions λ i are parameterized by LSTMs.

Original: To illustrate, as per (27), let k
Revised: To illustrate, as per (28), let k

Original: = 1 , set T i :
Revised: = 1 , set T i :

Original: = T i, 1 , and let A be an invertible matrix.
Revised: = T i, 1 , and let A be an invertible matrix.

Original: We can thenrecover Z related to the original Z ∗ as follows:
Revised: We can then recover Z related to the original Z ∗ as follows:

Original: [Equation]
Revised: [Equation]

Original: This means we can estimate the original latent variables up to point-wise transformations T ∗ i , T i .
Revised: This means we can estimate the original latent variables up to point-wise transformations T ∗ i , T i .

Original: In certain cases of non-sequential data, having A as a permutation matrix reduces the problem ofindeterminacy of Z ∗ to finding the point-wise transformations of Z. This is due to Eq.
Revised: In certain cases of non-sequential data, having A as a permutation matrix reduces the problem of indeterminacy of Z ∗ to finding the point-wise transformations of Z. This is due to Eq.

Original: (6) thenbecoming T ∗ i
Revised: (4) then becoming T ∗ i

Original: ( Z ∗ i )
Revised: ( Z ∗ i )

Original: =
Revised: =

Original: T i ′
Revised: T i ′

Original: ( Z i ′ )
Revised: ( Z i ′ )

Original: for a permuted index i ′ .
Revised: for a permuted index i ′ .

Original: Our deep latent variable model represented through parameters θ =
Revised: Our deep latent variable model represented through parameters θ =

Original: ( f, T i , λ i ) , as per (27), isp [Equation]
Revised: ( f, T i , λ i ) , as per (28), is p θ ( Y, W, Z | X ) =

Original:  , for a function f parameterized by an LSTM and sufficient statistic T i with parameters
Revised: p f ( Y, W | Z ) p T i ,λ i ( Z | X ) , for a function f parameterized by an LSTM and sufficient statistic T i with parameters

Original: λ i , i ∈{ 1 , . . .
Revised: λ i , i ∈ { 1 , . . .

Original: , N } .
Revised: , N } .

Original: Some of the identifiability assumptions for a VAE as described in (27) are that functionf is injective, and that sufficient statistic T i is differentiable almost everywhere.
Revised: Some of the identifiability assumptions for a VAE as described in (28) are that function f is injective, and that sufficient statistic T i is differentiable almost everywhere.

Original: Since our model isin principle a VAE, we rely on these assumptions and generate the synthetic data accordingly.
Revised: Since our model is in principle a VAE, we rely on these assumptions and generate the synthetic data accordingly.

Original: Wenote, however, that in a dynamical-system setting this might not always suffice for the causal modelto be identifiable and further theoretical work in this direction is required.
Revised: We note, however, that in a dynamical-system setting this might not always suffice for the causal model to be identifiable and further theoretical work in this direction is required.

Original: 3.3 Sequential CEVAE
Revised: 3.3 Sequential CEVAE

Original: Now we will introduce the Sequential Causal Effect Autoencoder (SCEVAE) (2).
Revised: We propose a novel method for time series causal link estimation under hidden confounding.

Original: It is based on CEVAE (3) in which all distributions involved in the model and inference networks are modeledby LSTMs (29).
Revised: It is predicated on CEVAE, but differs significantly in the architecture and relies on LSTMs to model the involved probability distributions.

Original: Furthermore, it models p ( Y | W, X ) sequentially, and introduces branching inthe decoder as depicted on the right of Fig.
Revised: Namely, we do not use a TARnet to estimate p ( Y | X, w 0 ) and p ( Y | X, w 1 ) , specialized for binary treatment, but rather model p ( Y | X, W ) sequentially, and introduce branching in the decoder as depicted on the right of Fig.

Original: This branching allows it to compare factual andcounterfactual causal effects, in order to estimate the cause-effect intensity after intervention on thenon-binary sequential cause variable.
Revised: This branching allows us to compare factual and counterfactual causal effects, in order to estimate the cause-effect intensity after intervention on the non-binary sequential cause variable.

Original: The encoder of SCEVAE is depicted in Fig.

Original: 2 on the left.

Original: We will now introduce the notation for sequential data and explain SCEVAE in more detail.
Revised: We will now introduce the notation for sequential data and explain the proposed method in full detail.

Original: Let X = { x t } Nt =1 , W = { w t } Nt =1 , and Y = { y t } Nt =1 be the time series of proxy, cause variable,and effect , for N ∈ N , respectively.
Revised: Let X = { x t } Nt =1 , W = { w t } Nt =1 , and Y = { y t } Nt =1 be the time series of proxy, cause variable, and effect, for N ∈ N , respectively.

Original: The hidden confounder Z = { z t } Nt =1 is also time series foreach z t ∈ R d , where d is the dimension of Z in the latent space at each time step t .
Revised: The hidden confounder Z = { z t } Nt =1 is also time series for each z t ∈ R d , where d is the dimension of Z in the latent space at each time step t .

Original: The conditionaldistributions of these variables used by SCEVAE framework are depicted in Fig.
Revised: The conditional distributions of these variables used by our method’s framework are depicted in Fig.

Original: The LSTMs aredenoted by f i , and each is parameterized by its own parameters ϕ
Revised: The LSTMs are denoted by f i , and each is parameterized by its own parameters ϕ

Original: i , for i ∈ { 1 , 2 , 3 , 4 } .
Revised: i , for i ∈ { 1 , 2 , 3 , 4 } .

Original: For each timestep t , we model x t , w t , and the prior of z t as follows:[Equation] [Equation] [Equation]
Revised: For each time step t , we model x t , w t , and the prior of z t as follows:

Original: [Equation]
Revised: [Equation]

Original: This means that each variable is full time series, since each individual time stepis modeled by a
Revised: This means that each variable is true time series, having each individual time step modeled by a

Original: Gaussian distribution.
Revised: Gaussian distribution.

Original: The LSTMs are not bidirectional, hence they do not have access to futurevalues of the input variable.
Revised: The LSTMs are not bidirectional, hence they do not have access to future values of the input variable.

Original: Using bidirectional LSTMs will be tackled in the future work.
Revised: This will be tackled in the future work.

Original: Whenthe proxy X contains binary components, SCEVAE models them according to p ( x t | z t ) =
Revised: When proxy X contains binary components, we model them as p ( x t | z t ) =

Original: Bern ( π = σ ( v ( z t ))) , for σ denoting a sigmoid function, and v : R −→ R , a real-valued functionparameterized by an LSTM.
Revised: Bern ( π = σ ( v ( z t ))) , for the sigmoid function σ , and v : R −→ R , a real-valued function parameterized by an LSTM.

Original: The mean and variance of the effect y t at each time step t are also parameterized by an LSTM:
Revised: The mean and variance of the effect y t at each time step t are also parameterized by an LSTM:

Original: [Equation]
Revised: [Equation]

Original: We indicate that in contrast to CEVAE, SCEVAE does not fix variance in Eq.
Revised: We indicate that in contrast to CEVAE, we do not fix variance in Eq.

Original: (10) but rather learn itfrom observational data.
Revised: (8) but rather learn it from observational data.

Original: Since W is continuous and a time series, SCEVAE models this distributionat each time step, thus avoiding branching our method’s architecture for different values of W , incontrast to CEVAE framework. SCEVAE approach is less restrictive, and can therefore be appliedto a wider variety of real-world problems. 
Revised: Our approach is less restrictive in the sense of its compatibility with continuous, sequential cause variable, and can therefore be applied to a wider variety of real-world problems. We use a Knockoff (3) of W as intervention since it preserves the original distribution of the data.

Revised: This is of particular importance for creating counterfactuals because the trained neural network anticipates test and training data stemming from the same distribution.

Original: The intervention it applies to W is setting it to Gaussiannoise. In contrast, we use knockoffs (5) for this purpose, since they preserve the original distributionof the  data as described in the following Subsection.
Revised: For comparison, we also apply Gaussian noise as intervention on W which removes the link to the hidden confounder but does not preserve the original distribution of W . One limitation of our approach is that we assume that, within the processed window, data is stationary and sufficient to produce good Knockoffs.

Original: According to the DAG in Fig. 1, the posterior distribution of Z depends on X , Y , and W .
Revised: According to the DAG in Fig. 1, the posterior distribution of Z depends on X , Y , and W .

Original: SCEVAEthus approximates it by:
Revised: We thus approximate it by:

Original: [Equation]
Revised: [Equation]

Original: For estimating the parameters of the auxiliary distribution of W and Y in Eq.
Revised: To estimate the parameters of the auxiliary distribution of W and Y in Eq.

Original: (15), SCEVAE uses thefollowing:
Revised: (13), we use the following:

Original: [Equation]
Revised: [Equation]

Original: To make sure that variances are positive, SCEVAE uses a softplus activation function softplus ( u ) =
Revised: To insure that variances are positive, we apply a softplus activation function softplus ( u ) =

Original: ln (1 + e u ) , for u ∈ R  to the output of a given LSTM cell used to parameterize the variance.
Revised: ln (1 + e u ) , for u ∈ R , to the output of a given LSTM cell used to parameterize the variance.

Original: Theregularization lossis weighted by λ ∈ R in order to obtain more stable effect predictions, so thevariational lower bound becomes:
Revised: We weigh the regularization loss by λ ∈ R in order to obtain more stable effect predictions, so the variational lower bound involving all modeled distributions is:

Original: [Equation]
Revised: [Equation]

Revised: Similarly to CEVAE, since it is necessary to know the intervention assignment W together with its outcome Y before inferring the posterior distribution over Z , two auxiliary distributions are introduced, helping to predict w t and y t for new samples.

Original: The variational lower bound used by SCEVAE is then:
Revised: The variational lower bound used as an objective for both the inference and the model networks of our method is then:

Original: [Equation]
Revised: [Equation]

Original: for x ∗ t , w ∗ t , y ∗ t being the observed values for the input, intervention, and outcome variables in thetraining set.
Revised: for x ∗ t , w ∗ t , y ∗ t being the observed values for the input, intervention, and outcome variables in the training set.

Revised: Our method’s encoder and decoder are depicted in Fig.

Revised: We call this novel approach Sequential

Revised: Causal Effect Variational Autoencoder (SCEVAE, pronounced /see-VAE/).

Original: 3.4 Knockoffs
Revised: 3.4 Knockoffs

Original: The idea of knockoffs was introduced by Barber and Candès (5) and it originates from the field offalse discovery rate control in the setup of finding potential explanatory variables to an observedresponse variable.
Revised: The idea of Knockoffs was introduced by Barber and Candès (3) and it originates from the field of false discovery rate control in the setup of finding potential explanatory variables to an observed response variable.

Original: They are meant as a tool for estimating feature importance using conditionalindependence tests.
Revised: They are meant as a tool for estimating feature importance using conditional independence tests but have recently been applied as intervention variables for causal discovery (8).

Original: Let u = γ · f ( Q ) + η, be a predictive model for a vector of responses u ∈ R n , an arbitrarily complexfunction f of a knowndesign matrix Q ∈ R n × p of potentially explanatory variables ( Q 1 , . . .
Revised: Let u = γ · f ( Q ) + η be a predictive model for a vector of responses u ∈ R n , an arbitrarily complex function f of a known matrix Q ∈ R n × p of potentially explanatory variables ( Q 1 , . . .

Original: , Q p ) ,an unknown vector of coefficients γ , and a Gaussian noise term η ∼ N (0 , σ 2 I ) .
Revised: , Q p ) , an unknown vector of coefficients γ , and a Gaussian noise term η ∼ N (0 , σ 2 I ) .

Original: One can thengenerate a knockoff of any feature Q j ∈ Q by first constructing a Gram matrix Σ = Q T Q , once allthe features Q j are normalized, so that Σ j,j = || Q j || 22 = 1 for all j = 1 , . . .
Revised: One can then generate a Knockoff of any feature Q j ∈ Q by first constructing a Gram matrix Σ = Q T Q , once all the features Q j are normalized, so that Σ j,j = || Q j || 2 2 = 1 for all j = 1 , . . .

Original: , p , and enforcing thefollowing two conditions.
Revised: , p , and enforcing the following two conditions.

Original: First, the knockoffs of Q , denoted as ˜ Q =
Revised: First, the Knockoffs of Q , denoted as ˜ Q =

Original: ( ˜ Q 1 , . . .
Revised: ( ˜ Q 1 , . . .

Original: , ˜ Q p ) , are constructedto have the same covariance structure as Q by enforcing ˜ Q T ˜ Q = Σ and Q T ˜ Q = Σ − diag ( s ) , fora p -dimensional non-negative vector s .
Revised: , ˜ Q p ) , are constructed to have the same covariance structure as Q by enforcing ˜ Q T ˜ Q = Σ and Q T ˜ Q = Σ − diag ( s ) , for a p -dimensional non-negative vector s .

Original: Moreover, the inter-variable correlations between differentoriginal and knockoff variables are enforced to be the same as those between the originals, that is,
Revised: Moreover, the inter-variable correlations between different original and Knockoff variables are enforced to be the same as those between the originals, that is,

Original: Q Tj ˜ Q k = Q Tj Q k for all j ̸ = k .
Revised: Q Tj ˜ Q k = Q Tj Q k for all j ̸ = k .

Original: We rely on the work by Romano et al.
Revised: We rely on the work by Romano et al.

Original: (6) to construct knockoffs for arbitrary and unspecifieddistributions of the observational data.
Revised: (6) to construct Knockoffs for arbitrary and unspecified distributions of the observational data.

Original: We chose the semidefinite programming (SDP) knockoffswhich are selected so that the original data and its knockoff are as decorrelated as possible.
Revised: We chose the semi-definite programming (SDP) Knockoffs which are selected so that the original data and its Knockoff are as decorrelated as possible.

Original:  Data 
Revised:  Data 4.1 Synthetic data

Original: Causal link estimation is impossible to validate unless we know the underlying system dynamics.
Revised: Causal link estimation is impossible to validate unless we know the underlying system dynamics.

Original: Forthis reason we use the following synthetic data to validate our approach with knockoff interventions.
Revised: For this reason we generate synthetic data.

Original: 4.1 Synthetic data
Revised: It contains a nonlinear causal link, all variables abide by the relations from DAG in Fig.

Original: We create synthetic data with a nonlinear causal link according to the following equations.
Revised: 1 and form the structural causal model according to the following equations.

Original: By z t , x t ,w t , ˆ w t , and y t we denote the hidden confounder, its proxy, cause variable,knockoff ˜ w t interventionof the cause variable, and the effect variable at time step t , respectively.
Revised: By z t , x t , w t , ˆ w t , and y t we denote the hidden confounder, its proxy, cause variable, and the effect variable at time step t , respectively.

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation] [Equation]
Revised: In the case of Knockoff intervention ˜ w t , the cause variable becomes ˆ w t = do ( w t = ˜ w t ) , and in the

Original: [Equation]
Revised: Gaussian noise intervention case we have ˆ w t ∼ N (0 , σ 2 ) , for σ 2 ∈ R .

Original: To inspect how our approach behaves when causal effects are linear, we modify Eq.
Revised: To asses how our method performs for estimating linear causal effects, we alter Eq.

Original: (20) as follows:
Revised: (17) as follows:

Original: [Equation]
Revised: [Equation]

Original: · w t + ϵ 4 .

Original: In Eqs.
Revised: In Eqs.

Original: (16)-(21), a,β, c i , d i , g, h, s, σ 2, σ 2 j ∈ R , i = 1 , 2 , τ ∈ N , and noise terms ϵ j = N (0 , σ 2 j ) ,for j = 1 , 2 , 3 , 4 , are mutually independent.
Revised: (14)-(18), a, c i , d i , g, h, s, σ 2 j ∈ R , i = 1 , 2 , τ ∈ N , and noise terms ϵ j = N (0 , σ 2 j ) , for j = 1 , 2 , 3 , 4 , are mutually independent.

Revised: By e we denote the Euler’s number.

Original: The time-varying proxy parameter b t  isdepicted in Fig.4 (in Appendix) and generated by setting b t = 2 N − 2 t for t = 1 , . . .
Revised: The time-varying proxy parameter b t (Fig. 4, Appendix A) is generated by setting b t = 2 N − 2 t for t = 1 , . . .

Original: , N 2 − 1 and symmetrically settingb N − t − 1 :
Revised: , N 2 − 1 and symmetrically b N − t − 1 :=

Original: = b t −1 for t = N 2 , . . .
Revised: b t −

Original: , N .
Revised: N .

Original: The counterfactual outcome ˆ y t for data with nonlinear and linearcausal link is obtained by substituting w t by ˆ w t in Eq.
Revised: The counterfactual outcome ˆ y t for data with nonlinear and linear causal link is obtained by substituting w t by ˆ w t in Eq.

Original: (20) and Eq.
Revised: (17) and Eq.

Original: (21), respectively.
Revised: (18), respectively.

Original: The parameters used in our experiments are, as per (2), a = 0 .
Revised: The parameters used in our experiments are a = 0 .

Original: 88 , β = 0 .
Revised: 88 , s = 0 .

Original: 5 ,s = 0 . 5 , τ = 100 ,c 1 = 0 .
Revised: 5 , τ = 100 , c 1 = 0 .

Original: 6 , c 2 = 0 .
Revised: 6 , c 2 = 0 .

Original: 2 , d 1 = 0 .
Revised: 2 , d 1 = 0 .

Original: 4 , d 2 = 0 .
Revised: 4 , d 2 = 0 .

Original: 8 , g = 0 .
Revised: 8 , g = 0 .

Original: 5 , h = 0 .
Revised: 5 , h = 0 .

Original: 5 , σ 2 = 0 . 6 , σ 22 = 1 , and σ 21 = 0 .
Revised: 5 , σ 2 = 0 . 6 , σ 22 = 1 , and σ 21 = 0 .

Original: 5 ,σ 22 = 0 . 8 , σ 23 = 0 .
Revised: 5 , σ 22 = 0 . 8 , σ 23 = 0 .

Original: 7 , and σ 24 = 0 .
Revised: 7 , and σ 24 = 0 .

Original: Furthermore, the hidden confounder and the cause variable areinitialized with z 0 = 0 .
Revised: The hidden confounder and the cause variable are initialized with z 0 = 0 .

Original: 1 and w 0 = 0 .
Revised: 1 and w 0 = 0 .

Original: 1 , whereas y 

Revised: 1 , whereas y

Original: = 0 .
Revised: = 0 .

Original: 4.2 Cloud and aerosol observation data
Revised: 4.2 Cloud and aerosol observation data

Original: We demonstrate our method’s applicability in real-world scenarios by applying it to cloud and aerosolobservations dataset provided by Jesson et al.
Revised: We demonstrate our method’s performance in real-world scenarios by applying it to cloud and aerosol observations dataset provided by Jesson et al.

Original: It consists of the 1 ◦ × 1 ◦ gridded version ofthe Moderate Resolution Imaging Spectroradiometer’s measurements obtained twice per day atapproximately 1 km × 1 km resolution from 2004 until 2019.
Revised: It consists of the 1 ◦ × 1 ◦ gridded version of the Moderate Resolution Imaging Spectroradiometer’s measurements obtained twice per day at approximately 1 km × 1 km resolution from 2004 until 2019.

Original: Data stems from the Pacific Basinregion off the coast of South America.
Revised: Data stems from the Pacific Basin region off the coast of South America.

Original: As per (31), we use the daily average of all variables todownsample.
Revised: As per (31), we use the daily average of all variables to downsample.

Original: As the outcome variable Y , we utilize cloud optical depth (COD) and as cause variable
Revised: As the outcome variable Y , we utilize cloud optical depth (COD) and as cause variable

Original: W the aerosol optical depth (AOD).
Revised: W the aerosol optical depth (AOD).

Original: In line with (31), we use the following meteorological variablesas proxies.
Revised: In line with (31), we use the following meteorological variables as proxies.

Original: Namely, sea surface temperature (SST), estimated inversion strength (EIS), verticalmotion at 500mb (w500), relative humidity at 700mb, 850mb, and 900mb (RH700, RH850, RH900).
Revised: Namely, sea surface temperature (SST), estimated inversion strength (EIS), vertical motion at 500mb (w500), relative humidity at 700mb, 850mb, and 900mb (RH700, RH850, RH900).

Original: As intervention, we use the knockoff of the cause variable AOD.
Revised: As intervention, we use the Knockoff of the cause variable AOD.

Original: All variables are normalized beforewe use them as input for training SCEVAE.
Revised: All variables are normalized before we use them as input for training SCEVAE.

Original:  Experiments
Revised:  Experiments

Original: Here we expound our experimental setup.
Revised: Here we expound our experimental setup.

Original: As per Trifunov et al.

Original: (2), we generate 1000 -time-stepslong multivariate time series and use the last 10% of each variable as test data without shuffling inorder to preserve inter-temporal dependencies.
Revised: We generate 1000 -time-step-long multivariate time series and use the last 10% of each variable as test data without shuffling in order to preserve inter-temporal dependencies.

Original: The last 10% of training data is used for validation.
Revised: The last 10% of training data is used for validation.

Original: We normalize the data by subtracting the mean of each variable and then dividing it by its standarddeviation.
Revised: We normalize the data by subtracting the mean of each variable and then dividing it by its standard deviation.

Original: We use batch size of 100 and randomly select that many consecutive time steps as batchesof the training data.
Revised: We use batch size of 100 and randomly select that many consecutive time steps as batches of the training data.

Original: The optimizer used is Adam (32) with learning rate of 10 − 5 , and a weightdecay of 10 − 3 .
Revised: The optimizer used is Adam (32) with learning rate of 10 − 5 , and a weight decay of 10 − 3 .

Original: We model the hidden confounder as five-dimensional in the latent space in all butone experiment where we explicitly investigate the influence of its dimensionality to causal linkestimation.
Revised: We model the hidden confounder as five-dimensional in the latent space in all but one experiment where we explicitly investigate the influence of its dimensionality to causal link estimation.

Original: The LSTMs of SCEVAE consist of two LSTM cells i.e. layers.
Revised: The LSTMs of SCEVAE consist of two LSTM cells i.e. layers.

Original: Since the LSTMs that weuse are not bidirectional, they can only look at the past time steps of a given time series.
Revised: Since the LSTMs that we use are not bidirectional, they can only look at the past time steps of a given time series.

Original: The hiddenstates and cell states of each LSTM have 32 dimensions.
Revised: The hidden states and cell states of each LSTM have 32 dimensions.

Original: We set the regularization parameter λ from 
Revised: We set the regularization parameter λ from Eq.

Original: Eq.
Revised: (12) to 0 . For all experiments, we used GPUs of type GeForce RTX 2080 Ti.

Original: 5.1 Results5.1.1 Synthetic data experiments
Revised: 5.1 Results 5.1.1 Synthetic data experiments

Original: To demonstrate the efficacy of our approach by comparing its results to the ground truth causal linkintensity, we apply it to synthetic data with either linear or nonlinear causal link betweenW and
Revised: To demonstrate the efficacy of our proposed method by comparing its results to the ground truth causal link intensity, we apply it to synthetic data with either linear or nonlinear causal link between

Original:  Y .
Revised: W and Y .

Original: Our method’s reconstruction of the effect variable Y is shown in Fig. 3 on the left, whereasthe reconstruction of the counterfactual effect variable ˆ Y is depicted in Fig.3 on the right.
Revised: Our method’s reconstruction of the effect variable Y and its counterfactual is ˆ Y is depicted in Fig.

Original: Wequalitatively observe that the reconstructions (orange) are quite close to the corresponding observedvalues (blue), accept in the case of outliers.
Revised: We qualitatively observe that the reconstructions (orange) are closely comparable to the corresponding observed values (blue).

Original: These variables are generated according to Eq.
Revised: These variables are generated according to Eq.

Original: (20) using W and (cid:99) W , respectively.
Revised: (17) using W and (cid:99) W , respectively.

Original: The quantitative results of SCEVAE’s causal analysis with and without knockoff interventions areshown in Table 1.
Revised: The quantitative results of SCEVAE’s causal analysis are shown in Table 1.

Original: The causality scores used are  ITE with a slight abuse of notation, denoting thedifference between the ground truth and the predicted ITE as per Eq.
Revised: The causality scores used are RMSE ITE , denoting the RMSE between the ground truth and the predicted ITE as per Eq.

Original: (1), factual Root Mean Squared Error (RMSE), and counterfactual RMSE.
Revised: (1), factual RMSE Y , and counterfactual RMSE ˆ Y .

Original: The latter two scores measure the discrepancy between theobserved Y and the estimated E ( Y | W, Z ) at each time step t , and the discrepancy between ˆ Y and E ( Y | (cid:99) W, Z ) at each time step t , respectively.
Revised: The latter two scores measure the discrepancy between the observed Y and the estimated E ( Y | W, Z ) at each time step t , and the discrepancy between ˆ Y and E ( Y | (cid:99) W, Z ) at each time step t , respectively.

Original: In the linear case, as per Table 1, the estimated causallink using SCEVAE with knockoffs compared to the ground truth is ITE = 0 . 44 when proxy parameterb is time-varying and ITE = 0 . 39 for a fixed value of parameter b = 0 . In the nonlinear case, fora time-varying parameter b ITE = 0 . 5 , which is quite lower than in thecase of SCEVAE withoutknockoff where
Revised: By "SCEVAE" in Table 1, we denote our method with Gaussian noise intervention on W , whereas "SCEVAE-Knockoff" indicates that we used the

Original: ITE = 0 . 66 . For a fixed b = 0 . 95 we obtain ITE = 0 . We compare our results onsynthetic data to those of Time Series Deconfounder (TSdeconf) (18) either without taking hiddenconfounding into account or when the substitutes for the hidden confounder are generated and usedas proxies.
Revised: Knockoff of W as intervention (cid:99) W .

Original: We note that TSdeconf only outputs RMSE between the ground truth and the predictedoutcome, but we generate the counterfactual RMSE by using ˆ w t and ˆ y t for training. Furthermore,since TSdeconf is not suitable for long time series, we generate 100 100 -time-steps-long trainingsamples for fairness. This is due to the fact that SCEVAE is trained on 100 100 -long epochs randomlychosen from our 1000 -long sequential variables. We note that in the case of the nonlinear causal link, TSdeconf performs slightly better than our
Revised: [Table]

Original: SCEVAEknockoff approach but with drastically higher variance, making our method much morestable. To obtain more stable results using TSdeconf, one would need to use much more trainingdata. In the case of the linear causal link, our methods superior performance becomes clearer for bothtime-varying and fixed proxy parameter b = 0 . [Table]
Revised: SCEVAE

Original: [Table]
Revised: [Table]

Revised: 59 ± 0.

Original: 0.5 ±
Revised: 0.79 ±

Original: [Table]
Revised: [Table]

Original: 0.77 ± 0.

Original: 0.72 ± 0.

Original: [Table]
Revised: [Table]

Original: 0.66 ±
Revised: 01 ±

Original: [Table]
Revised: [Table]

Original: 0.73 ± 0. 0.96 ± 0. TSdeconf (without) / 0.74 ± 0.
Revised: [Table]

Original: 0.57 ± 0. TSdeconf (with) / 0.74 ± 0. 0.54 ± 0.
Revised: [Table]

Original: [Table]
Revised: [Table]

Original: -knockoff 0.82 ± 0. 0.78 ± 0. 1 ± 0.
Revised: [Table]

Original: TSdeconf (without) / 0.74 ± 0.
Revised: [Table] ± [Table] [Table]

Original: 0.53 ± 0.
Revised: 7 ± [Table] TSdeconf (without) / 0.84 ± 0. 0.98 ± 0.

Original: TSdeconf (with) / 0.74 ± 0.
Revised: TSdeconf (with) / 0.84 ± 0.

Original: 0.58 ± 0.
Revised: 0.98 ± 0.

Original: The counterfactual RMSE metric shows that SCEVAE with knockoff intervention generates bettercounterfactual forecast than SCEVAE with standard normal intervention.
Revised: [Table] According to the results from Table 1, SCEVAE performs better in the linear than in the nonlinear causal link case, which illustrates the higher complexity of estimating nonlinear causal links.

Original:  This is expected since
Revised: We note that the methods works equally well for both fixed and time-varying values of the parameter b . Furthermore, in Table 1, we can clearly see how using Knockoffs improves the causal link intensity estimation by remarkably lower RMSE ITE metric, as well as by inducing lower variance, making this a more reliable intervention choice. In addition, it is especially interesting to note that the counterfactual RMSE is much lower in comparison to other methods which do not use Knockoff interventions. This is due to the fact that Knockoff (cid:99) W has the same distribution as W , and therefore produces a good reconstruction of the counterfactual outcome ˆ Y when we sample it from the learned estimate of p ( Y | W, Z ) .

Revised: We compare our results on synthetic data to those of Time Series Deconfounder (TSdeconf) (20) either without taking hidden confounding into account or when the substitutes for the hidden confounder are generated and used as proxies, in Table 1 denoted by "TSdeconf (without)" and "TSdeconf (with)", respectively.

Original: P ( W | Z ) is more similar to P ( (cid:99) W | Z ) in the case when (cid:99) Wis a knockoff of W , sharing its distribution.
Revised: We set the confounding parameter of TSdeconf to γ = 0 . 8 , and the number of substitute, as well as the simulated hidden confounders to one. We note that TSdeconf only outputs RMSE between the ground truth and the predicted factual or counterfactual outcome, so these are the main comparison metrics. For obtaining the factual RMSE T via TSdeconf we input X as covariates, W as treatment, and Y as outcome. To obtain the counterfactual RMSE ˆ Y , we also use X as covariates, but (cid:99) W , and ˆ Y as treatment and outcome, respectively.

Original: One can also note that SCEVAE with knockoff intervention performs better when parameter b is nottime-varying in the nonlinear case, but that the opposite is true in the  linear case.
Revised: Furthermore, since TSdeconf is not suitable for long time series, we generate 100 100 -time-step-long training samples for fairness. This is due to the fact that SCEVAE is trained on 100 100 -long epochs randomly chosen from our 1000 -time-step-long sequential variables. We note that in the case of the linear causal link, TSdeconf performs almost as well as our method but with much higher variance, making our method considerably more stable. To obtain stable results using TSdeconf, one would need much larger amount of training data.

Revised: In the case of the nonlinear causal link, our method’s superior performance becomes even clearer for both time-varying, and fixed proxy parameter b = 0 .

Revised: The factual RMSE T of the outcome reconstruction during training and test can be found in Fig.(Appendix A).

Original: In Fig. 6 in Appendix we show how dimensionality of Z in the latent space and the confoundingcoefficient d 2 influence the factual and counterfactual outcome’s reconstruction.
Revised: In Fig. 6 (Appendix A) we show how dimensionality of Z in the latent space and the confounding coefficient d 2 influence the factual and counterfactual outcome’s reconstruction.

Original: We tested latentdimensions D Z ∈ { 1 , 5 , 10 , 20 } , and coefficients d 2 ∈ { 0 . 8 , 1 , 1 . 2  } .
Revised: We tested latent dimensions D Z ∈ { 1 , 5 , 10 , 20 } , and coefficients d 2 ∈ { 0 . 8 , 1 , 1 . 2 , 1 . 6 } .

Original: We observe that RMSE valuesin both factual and counterfactual cases are lowest when d 2 = 0 . 8 and tend to increase as the rateof hidden confounding d 2 increases regardless of D Z .
Revised: We observe that RMSE values in both factual and counterfactual cases are lowest when d 2 = 0 . 8 and have an upward trend as the rate of hidden confounding d 2 increases regardless of D Z .

Original: As the dimensionality of Z increases, theuncertainty of the prediction becomes higher.
Revised: As the dimensionality of Z increases, the uncertainty of the prediction becomes higher.

Original: The factual reconstruction RMSE during training and test can be found in Appendix Fig.

Original: We notethat by increasing the number of latent dimensions the standard deviation of the prediction increases.

Original: 5.1.2 Cloud and aerosol data experiments
Revised: 5.1.2 Cloud and aerosol data experiments

Original: In the experiments on real aerosol-cloud-climate observations,  we demonstrate the importance ofchoosing good proxy variables and how SCEVAE with knockoff intervention improves the causallink estimation results.
Revised: In the experiments on real aerosol-cloud-climate observations, where we use COD as the effect and AOD as the cause variable, we demonstrate the importance of choosing suitable proxies.

Original: Since now we do not have the ground truth causal link intensity values, wecannot use the ITE error.
Revised: Since now we do not have the ground truth causal link intensity values, we cannot use RMSE ITE .

Original: Instead, we use ATE  of the predicted factual and counterfactual outcomevariables Y and ˆ Y .
Revised: Instead, we use ATE metric as per Eq. (2) of the predicted factual and counterfactual outcome variables Y and ˆ Y , respectively.

Original: In all experiments from Table 2 we use COD as outcome and AOD as intervention [Table] 309 variables, but  we vary the choice of proxy variable X .
Revised: Moreover, we note that intervention on real data is often not feasible in practice, but that by our method’s way of intervention we are able to counterfactually analyse the desired causal link.

Original: In case of meteorological proxy, we set it to bemultivariate including SST, EIS, w500, RH700, RH850, and RH900 as per (31).
Revised: In case of meteorological proxy, we set it to be multivariate including SST, EIS, w500, RH700, RH850, and RH900 as per (31).

Original: In theexperimentswhere proxy is indicated as uniform , we set it univariately touniform U (0 , 1) noise.
Revised: Whereas when the proxy is indicated as uniform , we set it univariately to U (0 , 1) .

Revised: The results of varying the choice of proxy X are shown in Table 2 for SCEVAE with standard normal, as well as with Knockoff intervention.

Original: We note that using meteorological proxies yields  lower variance of both training and test ATE incontrast to the case where we used uniform noise as the proxy.
Revised: We note that using meteorological proxies yields predominantly lower variance of both training and test ATE in contrast to the case where we used uniform noise as the proxy.

Revised: Admittedly, in this case of real data, SCEVAE with Knockoff intervention does yield higher RMSE T with the use of meteorological proxies.

Revised: This could be due to our assumption that in a given training window data is stationary and suitable for generating a Knockoff which might not always be the case.

Original: Moreover, we note that the true causallink between COD and AOD measured by ATE is stronger when we use SCEVAE with knockoffintervention than without it.
Revised: Furthermore, we note that the true causal link between COD and AOD measured by ATE is stronger when we use SCEVAE with Knockoff than when it is used with Gaussian noise intervention.

Original: Furthermore, when using knockoff intervention the causal link intensitybetween COD and AOD is stronger during training when meteorological proxies are employed thanin the case of setting X to uniform noise.
Revised: Furthermore, when using Knockoff intervention the causal link intensity between COD and AOD is stronger during training when meteorological proxies are employed, than in the case of setting X to uniform noise.

Original: This is opposite during test.
Revised: This is opposite during test but still reflects the unsuitable use of uniform noise as the proxy by exhibiting very high ATE variance.

Original:  Conclusion
Revised:  Conclusion

Original: Causal link intensity estimation is a challenging task, especially in the presence of latent confounders.
Revised: Causal link intensity estimation is a challenging task, especially in the presence of latent confounders.

Original: In this paper we proposed to use SCEVAE with knockoff intervention for  this purpose. We achievedmore stable results than the  time series deconfounding benchmark TSdeconf and were able toestimate the confounded causal link intensity more accurately than SCEVAE with standard normalinterventions. This was attained on synthetic data with both linear and nonlinear causal links.
Revised: In this paper we introduced SCEVAE, a novel deep learning method for time series causality analysis under hidden confounding to tackle this problem in sequential data with Knockoff interventions. It is inspired by the CEVAE framework, but applicable to complex non-stationary time series through the use of LSTMs and fundamental architectural novelty.

Original: In addition, we observed that using knockoff interventions reduces the counterfactual RMSE incomparison to SCEVAE without knockoff intervention since the distribution of the knockoff agreeswith the learned distribution of the cause variable. Moreover, through the experiments on realcloud-aerosol observational data, we indicated our method’s applicability to real-world problems andillustrated how the use of proxy contributes to better causal link intensity estimation.
Revised: Our method allows for single-variable causality analysis instead of using many independent samples for training.

Original: For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope? [Yes](
Revised: We achieved better and more stable results than the time series deconfounding benchmark TSdeconf.

Original: b

Original: ) Did you describe the limitations of your work Yes] c ) Did you discuss any potential negative societal impacts of your work
Revised: Moreover, we showed that estimating the confounded causal link intensity can be done more accurately with Knockoff rather than using standard normal interventions.

Original: [N/A]

Original: d

Original: ) Have you read the ethics review guidelines and ensured that your paper conforms tothem

Original: [Yes If you are including theoretical results... (a) Did you state the full set of assumptions of all theoretical results? [N/A]
Revised: This was attained on synthetic data with both linear and nonlinear causal links.

Original: (b) Did you include complete proofs of all theoretical results? [N/A] If you ran experiments... (a) Did you include the code, data, and instructions needed to reproduce the main experi-
Revised: In addition, we observed that using Knockoff interventions reduces the counterfactual RMSE

Original: mental results (either in the supplemental material or as a URL)?

Original: [No] We have not included the code, but we provided the equations through which one can recreate the synthetic data, and the real data we used is publicly available. The instructions to
Revised: ˆ Y in comparison to SCEVAE without Knockoff intervention since the distribution of the Knockoff agrees with the learned distribution of the cause variable.

Original: reproduce the main experimental results are provided in the text in Section 3.3 andSection 5. (b) Did you specify all the training details (e g., data splits, hyperparameters, how theywere chosen)? [Yes]
Revised: Moreover, through the experiments on real cloud-aerosol observational data, we indicated our method’s potential applicability to real-world problems and illustrated how the use of meaningful proxies contributes to its identifiability

Original: (c) Did you report error bars (e

Original: g., with respect to the random seed after running experiments multiple times)?

Original: [Yes] (d) Did you include the total amount of compute and the type of resources used (e g., typeof GPUs, internal cluster, or cloud provider)? [Yes]
Revised: Acknowledgements

Original: If you are using existing assets (e g., code, data, models) or curating/releasing new assets... (a) If your work uses existing assets, did you cite the creators? [Yes]
Revised: Funding for this study was provided by the German Research Foundation (DFG) individual research grant SH 1682/1-1 and the European Research Council (ERC) Synergy Grant “Understanding and

Original: (b) Did you mention the license of the assets?

Original: [N/A]

Original: (c) Did you include any new assets either in the supplemental material or as a URL?

Original: [Yes]

Original: We included a new synthetic dataset model. (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
Revised: Modelling the Earth System with Machine Learning (USMILE)” under the Horizon 2020 research and innovation programme (Grant agreement No. 855187 ).

Original: (e) Did you discuss whether the data you are using/curating contains personally identifiable

Original: information or offensive content?

Original: [N/A]

Original: If you used crowdsourcing or conducted research with human subjects...

Original: (a) Did you include the full text of instructions given to participants and screenshots, ifapplicable?

Original: [N/A]

Original: (b) Did you describe any potential participant risks, with links to Institutional Review

Original: Board (IRB) approvals, if applicable?

Original: [N/A]

Original: (c) Did you include the estimated

Original: hourly wage paid to participants and the total amountspent on participant compensation?

Original: [N/A]

Original: A Appendix
Revised: A Appendix