Original: Causal Discovery using Model Invariance through Knockoff Interventions
Revised: Causal Discovery using Model Invariance through Knockoff Interventions

Original: Abstract
Revised: Abstract

Original: Cause-effect analysis is crucial to understand the underlying mechanism of a system.
Revised: Cause-effect analysis is crucial to understand the underlying mechanism of a system.

Original: We propose to exploit model invariance through interventions on the predictors to infer causality in a non-linear multivariate system of time series.
Revised: We propose to exploit model invariance through interventions on the predictors to infer causality in nonlinear multivariate systems of time series.

Original: We model nonlinear interaction in time series using DeepAR and then expose the model to different environments using knockoff intervention to test model invariance.
Revised: We model nonlinear interactions in time series using DeepAR and then expose the model to different environments using Knockoffs-based interventions to test model invariance.

Original: Knockoffs are  in-distribution  null variables generated without knowing the response.
Revised: Knockoff samples are pairwise exchangeable, in-distribution and statistically null variables generated without knowing the response.

Original: We test model invariance where we show that the distribution of the response residual does not change significantly upon interventions on non-causal features.
Revised: We test model invariance where we show that the distribution of the response residual does not change significantly upon interventions on non-causal predictors.

Original: We use synthetically generated time series to evaluate and compare our approach with other causality methods. Overall ourproposed method outperforms other widely used methods.
Revised: We evaluate our method on real and synthetically generated time series. Overall our method outperforms other widely used causality methods, i.e, VAR Granger causality, VARLiNGAM and PCMCI + .

Original: 1. Introduction
Revised: 1. Introduction

Original: Discovering causal graph in a system is vital to understand system’s dynamics and its response to changes in environment.
Revised: Discovering causal graph in a system is vital to understand system's dynamics and its response to changes in environment.

Original: The methods which extract causal information from observational data are crucial for the broad field of artificial intelligence (AI) (Jordan, 2019).
Revised: The methods which extract causal information from observational data are crucial for the broad field of artificial intelligence.

Original: In this work we use deep networks to learn nonlinear interactions in multivariate time series and test model invariance through knockoff interventions for causal inference.
Revised: In this work we use deep networks to learn nonlinear interactions in multivariate time series and test model invariance through Knockoffs (Barber & Cand`es, 2015; Candes et al., 2017) based interventions for causal inference.

Original: We build our work on the assumption that the causal mechanism in a real-world system does not change across diverse environments (Peters et al., 2016; Heinze-Deml et al., 2018; Pfister et al., 2019).
Revised: We build our work on the assumption that the causal mechanism in a real-world system does not change across diverse environments (Peters et al., 2016; HeinzeDeml et al., 2018; Pfister et al., 2019).

Original: More specifically, the conditional distribution of the response given its causal predictors remains invariant for all observations over different settings.
Revised: More specifically, the conditional distribution of the response given its causal predictors remains invariant for all observations over different settings.

Revised: A model is regarded as invariant if the distribution of the residuals of model's response does not change when intervened on its non-causal predictors.

Original: Deep networks best suit to learn nonlinear interactions in a multivariate system.
Revised: Deep networks best suit to learn nonlinear interactions in a multivariate system.

Original: We use DeepAR (Salinas et al., 2020) to model nonlinear relations in multivariate time series.
Revised: We use DeepAR (Salinas et al., 2020) to model nonlinear relations in multivariate time series, which is based on recurrent neural networks (RNNs) and generate probabilistic forecast.

Original: DeepAR characterizes conditioning on observed variables and estimate latent features  from time series that might help in the identification of spurious associations.
Revised: It characterizes conditioning on observed variables and estimate latent features i.e. trends and seasonality from time series that might help in the identification of spurious associations.

Original: To expose trained models to unseen settings for causal inference, we can intervene by omitting or replacing model predictors.
Revised: To expose trained models to unseen settings for causal inference, we can either omit or replace model predictors with a different representation.

Original: Since, deep networks can not handle missing input or outof-distribution (OOD) data, we propose to use the theoretically well-established Knockoffs framework(Barber et al., 2020) to generate interventional environments.
Revised: Since, deep networks cannot handle missing input or out-of-distribution (OOD) data, we propose to use the theoretically well-established Knockoffs framework to generate interventional environments.

Original: Knockoff copies are statistically null-variables, in-distribution and as uncorrelated as possible with the originals.
Revised: Knockoffs are statistically null-variables, in-distribution and as uncorrelated as possible with the originals.

Revised: Moreover,

Revised: Knockoffs possess the pairwise exchangeability property when means that replacing any variable with their knockoff copy in a multivariate setup will not change the underlying joint distribution of the set of variables.

Original: To highlight the effectiveness of Knockoffs in our method, we compare it with other types of interventions such asmean-distribution, OOD,  uniform distribution.
Revised: We highlight the effectiveness of Knockoffs as interventional variables by comparing it with other intervention methods such as OOD, mean and uniform distribution.

Original: Using our approach, we discover full causal graph assuming causal sufficiency and stationarity in  our data. We compare our work with VAR 
Revised: We discover full causal graph assuming causal sufficiency and stationarity in synthetic and real data and show that our method outperforms widely used causality methods, i.e., VAR Granger causality (VAR-GC) (Granger, 1969), VARLiNGAM

Original: Granger causality (VAR-GC), VARLiNGAM and PCMCI (Runge et al., 2019) using synthetic dataset.
Revised: (Hyv¨arinen et al., 2010), PCMCI + (Runge, 2020) as well as other applied intervention methods.

Original: We show that our method outperforms these methods in nonlinear causal discovery.

Original: The remaining of this paper is organized as follows.
Revised: The remaining of this paper is organized as follows.

Original: In
Revised: In

Original: Section 2, we cover the related work.
Revised: Section 2, we cover the related work.

Original: We provide methodological background in Section 3.
Revised: We provide methodological background in Section 3.

Original: In Section 4, we describe our proposed method.
Revised: In Section 4, we describe our proposed method.

Original: Results are presented and discussed in Section 5.
Revised: Results are presented and discussed in Section 5.

Original: In Section 6 we conclude our work.
Revised: In Section 6 we conclude our work.

Original: 2. Related Work
Revised: 2. Related Work

Original: The methods in (Peters et al., 2016; Heinze-Deml et al., 2018; Pfister et al., 2019; Gamella & Heinze-Deml, 2020;
Revised: The methods in (Peters et al., 2016; Heinze-Deml et al., 2018; Pfister et al., 2019; Gamella & Heinze-Deml, 2020;

Original: Martinet et al., 2021) use invariant causal prediction (ICP) for causal inference assuming that data samples are drawn from heterogeneous environments and can identify the ancestors of the response.
Revised: Martinet et al., 2021) use invariant causal prediction (ICP) for causal inference assuming that data samples are drawn from heterogeneous environments and can identify the ancestors of the response.

Original: These methods are limited in a sense that they estimate causal relationship of the candidate variable with the target only rather than discovering full causal graph as done in our work through multivariate target modelling.
Revised: These methods are limited in a sense that they estimate causal relationship of the candidate variable with the target only rather than discovering full causal graph as done in our work through multivariate target modelling.

Original: Moreover, they search for subset of predictors which remains unchanged throughout environments while we estimate a consistent causal set using model invariance across interventional environments over multiple forecast windows.
Revised: Moreover, they search for subset of predictors which remains unchanged throughout environments while we estimate a consistent causal set using model invariance across interventional environments over multiple forecast windows.

Original: ICPthen outputs the intersection of all subsets of invariant predictor which forms a strict subset of all direct causes or may even be empty.
Revised: ICP outputs the intersection of all subsets of invariant predictor which forms a strict subset of all direct causes or may even be empty.

Original: This is because disjoint sets of predictors can be invariant, yielding an empty intersection.
Revised: This is because disjoint sets of predictors can be invariant, yielding an empty intersection.

Original: In (Mogensen et al., 2022), the authors introduce and characterize minimally invariant sets of predictors, that is, invariant sets S for which no proper subset is invariant.
Revised: In (Mogensen et al., 2022), the authors introduce and characterize minimally invariant sets of predictors, that is, invariant sets S for which no proper subset is invariant.

Original: They propose to consider the union S IAS of all minimally invariant sets, where IAS stands for invariant ancestry search.
Revised: They propose to consider the union S IAS of all minimally invariant sets, where IAS stands for invariant ancestry search.

Original: The method in (Peters et al., 2016) works with nonstationary data assuming that the environments are known in order to exploit the invariance.
Revised: The method in (Peters et al., 2016) works with nonstationary data assuming that the environments are known in order to exploit the invariance.

Original: It is argued that without knowledge of the environments, the causal inference becomes difficult.
Revised: It is argued that without knowledge of the environments, the causal inference becomes difficult.

Original: Moreover, estimating the environments from data in naive manner i.e. clustering, change point detection, etc., and subsequently applying the existing methodology may lead to a loss of the methods coverage, guarantee or yield less powerful results.
Revised: Moreover, estimating the environments from data in naive manner e.g., clustering, change point detection, etc., and subsequently applying the existing methodology may lead to a loss of the methods coverage, guarantee or yield less powerful results.

Original: The authors of (Pfister et al., 2019) do not estimate the environments but instead utilizes the existing non-invariances in observations.
Revised: The authors of (Pfister et al., 2019) do not estimate the environments but instead utilizes the existing non-invariances in observations.

Original: In our method, we generate knockoff representation of the predictors to represent different environments and test model invariance.
Revised: In our method, we generate knockoff representation of the predictors to represent different environments and test model invariance for causal inference in nonlinear systems.

Original: The work in (Ahmad et al., 2021) uses deep networks with knockoff intervention to estimate Granger causality in nonlinear  time series.
Revised: Previously we used deep networks with Knockoffs-based interventions to estimate Granger causality in nonlinear multivariate time series (Ahmad et al., 2021).

Original: The authors of (Tank et al., 2021; Romano et al., 2020) uses deep networks combined with sparsity-inducing penalties on the weights to estimate causal significance of the predictors.
Revised: The authors of (Tank et al., 2021; Romano et al., 2020) utilize deep networks combined with sparsity-inducing penalties on the weights to estimate causal significance of the predictors.

Original: However, it is difficult to retrieve clear information from weights of the deep networks that can be used for interpretability (Luo et al., 2020).
Revised: However, it is difficult to retrieve clear information from weights of the deep networks that can be used for interpretability (Luo et al., 2020).

Original: Attention-based networks have also been used, where network parameters are utilized to determine time delay in the cause-effect relationship (Nauta et al., 2019).

Original: The

Original:  PCMCI (Runge et al., 2019) method uses linear and nonlinear conditional independent (CI) tests to infer causality in multivariate time series.
Revised: The widely used PCMCI (Runge et al., 2019) method applies linear and nonlinear conditional independent (CI) tests to infer causality in multivariate time series.

Original: The extended version of causeeffect variational autoencoders (CEVAE) (Louizos et al., 2017) integrates the domain-specific knowledge for nonlinear causal inference in ecological time series (Trifunov et al., 2019).
Revised: The extended version of cause-effect variational autoencoders (CEVAE) (Louizos et al., 2017) integrates the domain-specific knowledge for nonlinear causal inference in ecological time series (Trifunov et al., 2019).

Original: Time-frequencybased analysis has been done to handle spurious associations in time series (Shadaydeh et al., 2019). However, they use spectral representation of linear VAR-GC andhence might struggle in case of nonlinearity.
Revised: Time-frequency analysis has been done to handle spurious associations in time series (Shadaydeh et al., 2019) however, they use spectral representation of linear VAR-GC and might struggle in case of nonlinearity.

Original: 3. Methodological Background
Revised: 3. Methodological Background

Original: In this section we introduce the necessary background on
Revised: In this section we introduce the necessary background on

Original: DeepAR(Salinas et al., 2020) and the Knockoffs framework (Romano et al., 2020).
Revised: DeepAR and the Knockoffs framework.

Original: 3.1. DeepAR
Revised: 3.1. DeepAR

Original: DeepAR is a powerful method suitable for probabilistic forecasting of high dimensional nonlinear non-stationary time series.
Revised: DeepAR is a powerful method suitable for probabilistic forecasting of high dimensional nonlinear non-stationary time series (Salinas et al., 2020).

Original: It uses LSTM-based autoregressive recurrent neural networks (Graves, 2013) and learns nonlinear interactions from the history of all the related time-series.
Revised: It uses LSTM-based autoregressive RNNs (Graves, 2013) and learns nonlinear interactions from the history of all the related time-series.

Original: Few key attributes of DeepARcompared to other forecasting methods are: It extracts hidden features in the form ofseasonal patterns and trends  inthe data, and characterizes conditioning on the entire set of variables during inference.
Revised: Few key attributes of DeepAR are: It extracts hidden features in the form of trends and seasonality in data, and characterizes conditioning on the entire set of variables during inference which can be helpful in identifying spurious relations.

Original: Moreover,in this method, lesser hand-crafted feature engineering is needed  to capture complex, group-dependent behavior.
Revised: Moreover, lesser hand-crafted feature engineering is needed in DeepAR to capture complex, group-dependent behavior.

Original: 3.2. Knockoffs
Revised: 3.2. Knockoffs

Original: The Knockoffs framework is developed to estimate feature importance using CI testing with controllable false discovery rate (Barber & Cand`es, 2015).
Revised: The Knockoff framework is developed to estimate feature importance using CI testing with controllable false discovery rate (Barber & Cand ` es, 2015).

Original: Given a set of observed variables Z
Revised: Given a set of observed variables Z

Original: =
Revised: =

Original: ( Z 1 , . . .
Revised: ( Z 1 , . . .

Original: , Z n ) in an environment with known distribution P Z , and a predictive model, the knockoffs of the observed variables, defined as (cid:101)
Revised: , Z n ) in an environment with known distribution P Z , and a predictive model, the knockoff copies of the observed variables, defined as (cid:101)

Original: Z =
Revised: Z =

Original: ( (cid:101) Z 1 , . . .
Revised: ( (cid:101) Z 1 , . . .

Original: , (cid:101) Z n ) , represents different environments while maintaining the indistribution and uncorrelation property of the original environment variable.
Revised: , (cid:101) Z n ) , represent interventional environments while maintaining the in-distribution and decorrelation property of the observational environment.

Original: Moreover, Knockoffs are  null-variables,which means they are generated without looking at model response.
Revised: Knockoff samples are statistically null-variables, generated without knowing model response.

Original: Furthermore, knockoff variables should satisfy the pairwise exchangeability condition (Barber & Cand`es, 2015)
Revised: Moreover, knockoffs should satisfy the pairwise exchangeability condition (Barber & Cand ` es, 2015)

Original: [Equation]
Revised: [Equation]

Original: for any subset A ⊆ 1 , . . . , n , here d = represents equal distributions.
Revised: for any subset A ⊆ 1 , . . . , n , here d = represents equal distributions.

Original: The ( Z, (cid:101) Z ) swap ( A ) is obtained from ( Z, (cid:101) Z ) by swapping the entries Z j and (cid:102)
Revised: The ( Z, (cid:101) Z ) swap ( A ) is obtained from ( Z, (cid:101) Z ) by swapping the entries Z j and (cid:101)

Original: Z j for each j ∈ A .
Revised: Z j for each j ∈ A .

Original: The knockoff mechanism can be thought of as generating a probability distribution P (cid:101)
Revised: The knockoff mechanism can be thought of as generating a probability distribution P (cid:101)

Original: Z | Z ( .
Revised: Z | Z ( .

Original: | z ) which is the conditional distribution of (cid:101)
Revised: | z ) which is the conditional distribution of (cid:101)

Original: Z given Z =
Revised: Z given Z =

Original: z chosen such that the obtained joint distribution of ( Z, (cid:101) Z ) which is equal to
Revised: z chosen such that the obtained joint distribution of ( Z, (cid:101) Z ) being equal to

Original: [Equation] P 
Revised: P Z ( z ) P (cid:101)

Original: [Equation] ) , is pairwise ( z j , (cid:101)
Revised: Z | Z ( (cid:101) z | z ) , is pairwise ( z j , (cid:101)

Original: z j ) symmetric and satisfy the exchangeability condition in (1).
Revised: z j ) symmetric and satisfy the exchangeability condition in (1).

Original: For variables with Gaussian distribution
Revised: For variables with Gaussian distribution

Original: P Z = N n ( 0 n , Σ) , (Candes et al., 2017) shows that knockoffs (cid:101)
Revised: P Z = N n ( 0 n , Σ) , (Candes et al., 2017) shows that knockoffs (cid:101)

Original: Z i, ∗ can be drawn from the conditional distribution
Revised: Z i, ∗ can be drawn from the conditional distribution

Original: P 
Revised: P (cid:101) Z | Z ( . | Z i, ∗ ) = N n (( I n − S Σ − 1 )

Original: [Equation] for any fixed diagonal matrix S satisfying 0 ≤ S ≤ 2Σ .
Revised: Z i, ∗ , 2 S − S Σ − 1 S ) (2) for any fixed diagonal matrix S satisfying 0 ≤ S ≤ 2Σ .

Original: This leads tothe joint distribution of ( Z i, ∗ , (cid:101)
Revised: This leads to joint distribution of ( Z i, ∗ , (cid:101)

Original: Z i, ∗ ) being equal to
Revised: Z i, ∗ ) being equal to

Original: N [Equation]{+2 n (cid:18) 
+}
Revised: N {+2 n (cid:18) 
+}

Original:  , which satisfies the pairwise exchangeablity (1).
Revised: 2 n , (cid:20) Σ Σ − S Σ − S Σ (cid:21)(cid:19) , which satisfies the pairwise exchangeablity (1).

Original:  S  represents the difference between the original variable and their knockoff version and the goal is the maximize S so that the  term Σ − S is minimized which shows the covariance between the original and their knockoff representation.
Revised: Here S is to be maximized which represents the difference between the original variables and their knockoff version. As a result the off-diagonal term Σ − S is minimized making the covariance of the original variable with its corresponding knockoff version as low as possible.

Original: 4. Causality using Model Invariance through Knockoffs
Revised: 4. Causality using Model Invariance through Knockoffs

Original: Let z i , i = 1 , . . .
Revised: Let z i , i = 1 , . . .

Original: , N be the N − variate time series.
Revised: , N be the N − variate time series.

Original: Each time series z
Revised: Each time series z

Original: i,t , t = 1 , . . .
Revised: i,t , t = 1 , . . .

Original: , r is a realization of length r realvalued discrete stochastic process Z i , i = 1 , . . .
Revised: , r is a realization of length r realvalued discrete stochastic process Z i , i = 1 , . . .

Original: , N .
Revised: , N .

Original: Our goal is to estimate the causal relation among all variables and retrieve a full causal graph.
Revised: Our goal is to estimate the causal relations among all variables and retrieve a full causal graph.

Original: Throughout our work, we assume stationarity and causal sufficiency which implies
Revised: Throughout our work, we assume stationarity and causal sufficiency which implies

Original: Z i , i = 1 , . . .
Revised: Z i , i = 1 , . . .

Original: , N is a stationary stochastic process, and the set of observed variables Z i , i = 1 , . . .
Revised: , N is a stationary stochastic process, and the set of observed variables Z i , i = 1 , . . .

Original: , N includes all of the common causes of pairs in Z .
Revised: , N includes all of the common causes of pairs in Z .

Original:  We exploit  model invariance for causal discovery inmultivariate nonlinear time series where we determine that the underlying causal structure of a system remains invariant irrespective of the interventional environments.
Revised: An extension of the current work to deal with non-stationary settings is in progress. We exploit the idea of model invariance for causal discovery in nonlinear time series where we determine that the underlying causal structure of a system remains invariant over interventional environments.

Original: We model nonlinear time series with DeepAR and generate knockoff copies of the original data to test model invariance.
Revised: We model nonlinear time series with DeepAR and generate knockoff copies of the original data to test model invariance.

Original: The invariance of model is tested over multiple forecast windows so that we obtain a consistent causal graph.

Original: A model is regarded invariant if the distribution moments of the response residuals i.e µ or σ 2 do not change by intervening on the non-causal predictors.
Revised: A model is regarded invariant if the distribution, i.e., µ and σ 2 , of the response residuals given its causal predictors do not change against interventional environments.

Original: The variables are considered as non-causal if the residual distribution of the model response does not significantly change by exposing the model to their knockoff representation.
Revised: The predictors are considered as non-causal if the residual distribution of the model response does not significantly change by exposing it to their knockoff representations.

Original: We retrieve causal graph with a consistent set of causal predictors using hypothesis testing over residual distribution from multiple forecast windows.
Revised: We retrieve causal graph with a consistent set of variables using hypothesis testing over residual distribution from multiple forecast windows.

Original: Any type of distribution comparison test can be associated with our method.

Original: To generate knockoffs intervention data, we implement DeepKnockoffs (Romano et al., 2020), a framework for sampling approximate knockoffs.
Revised: To generate knockoff variables, we implement DeepKnockoffs (Romano et al., 2020), a framework for sampling approximate knockoffs.

Original: We generate knockoff copies using a mixture of Gaussian models as implemented in (Gimenez et al., 2019).
Revised: We use mixture of Gaussian models as implemented in (Gimenez et al., 2019) to obtain knockoff samples.

Original: The procedure consists of first sampling the mixture assignment variable from the posterior distribution.
Revised: The procedure consists of first sampling the mixture assignment variable from the posterior distribution.

Original: The knockoffs are then sampled from the conditional distribution of the knockoffs given the original variables and the sampled mixture assignment such that the necessary conditions for knockoffs are satisfied.
Revised: The knockoff variables are then sampled from the conditional distribution of the Knockoffs given the original variables and the sampled mixture assignment such that the necessary conditions for Knockoffs are satisfied.

Original: Swapping any knockoff variable with their original version must leave the underlying joint data distribution invariant as in 1.
Revised: Swapping any knockoff variable with their original version must leave the underlying joint data distribution invariant as in (1).

Original: For Gaussian knockoffs, we know that Z ∼ N ( µ, Σ) .
Revised: Given observed variable Z ∼ N ( µ, Σ) , we aim to build its knockoff copy (cid:101) Z in such a way that the joint distribution of

Original: We aim to build knockoffs in such a way that the joint distribution of ( Z, (cid:101) Z ) ∼ N ( ∗ , ∗∗ ) is Gaussian.
Revised: [Equation] ∼ N [Equation] ∗ , ∗∗ ) is Gaussian and

Original: [Equation]
Revised: [Equation]

Original: Where S is a diagonal matrix such that ∗∗ ⪰ 0 .
Revised: where S is a diagonal matrix such that ∗∗ ⪰ 0 .

Original: S represents the discrepancy between originals and knockoffs.
Revised: S represents the discrepancy between originals and knockoffs.

Original: We have the same µ for both originalvariable and knockoff sample.
Revised: We have the same µ for both original and knockoff version. The

Original: For covariance matrix, the diagonal entries show autocovariance while theoff diagonal value represent covariance  of  original variable with their knockoff version which should be minimized by maximizing S .
Revised: Knockoff framework maintain the same covariance among variables in set of generated Knockoffs as in originals which is represented by the diagonal entries in given covariance matrix. However, the covariance Σ − S of the original variable with its corresponding knockoff copy in the offdiagonals is kept as low as possible by maximizing S .

Original: We obtain knockoffs (cid:101)
Revised: We obtain Knockoffs (cid:101)

Original: Z j , j ⊆ 1 , · · · , n , for all variables in the multivariate time series and substitute one variable Z j at a time with (cid:101) Z j for each j ⊆ 1 , · · · , n to test model invariance and estimate cause effect relationship.
Revised: Z j , j ⊆ 1 , · · · , n , for all variables in the multivariate time series and substitute one variable Z j at a time with (cid:101) Z j for each j ⊆ 1 , · · · , n to test model invariance and estimate cause-effect relationships.

Original: To estimate causal effect of z i , i = 1 , . .
Revised: To estimate causal effect of z i , i = 1 , . .

Original: , N on the response z j , we test model invariance through its forecast residuals by feeding a knockoff representation (cid:101)
Revised: , N on the response z j , we test model invariance using its forecast residuals by feeding a knockoff representation (cid:101)

Original: z i of the original variable z i .
Revised: z i of the original variable z i .

Original: We obtain residuals e 1 , e 2 , ..., e n ∼ R and ˜ e 1 , ˜ e 1 , ..., ˜
Revised: We obtain residuals e 1 , e 2 , ..., e n ∼ R and ˜ e 1 , ˜ e 1 , ..., ˜

Original: e n ∼ ˜ R overmultiple forecast windows  for each realization z j of the stochastic process Z j , j = 1 , . . .
Revised: e n ∼ ˜ R over forecast windows w in range [20 − 30] with a step size of [5 − 10] for each realization z j,t of the stochastic process Z j , j = 1 , . . .

Original: , N with and without intervention on z i , i ̸
Revised: , N with and without intervention on z i , i ̸

Original: =
Revised: =

Original: j .
Revised: j .

Original: R represents distribution of model residuals for actual outcome and ˜ R shows residual distribution after intervention.
Revised: R represents distribution of model residuals for actual outcome and ˜ R shows residual distribution after intervention.

Original: We compare these two distributionsfor inferring causality between each two nodes using Kolmogorov–Smirnov  test (Smirnov, 1939).
Revised: We compare these two distributions using Kolmogorov–Smirnov (KS) test (Smirnov, 1939) for causal inference.

Original: The test statistic uses the supremum distance D n,m between R and ˜ R .
Revised: The test statistic uses the supremum distance D n,m between R and ˜ R .

Original: Here n, m are the number of samples in the these two distributions.
Revised: Here n, m are the number of samples in the these two distributions.

Original: [Equation]
Revised: [Equation]

Original: The null hypothesis H 0 :
Revised: The null hypothesis H 0 :

Original: z i does not cause z j , is accepted if the distribution of the residuals is approximately identical across environments
Revised: z i does not cause z j , is accepted if the distribution of the residuals is approximately identical across environments

Original: ˜ R d ≈ R i with a significant level α = [5% ∼ 10%] .
Revised: ˜ R d ≈ R i with a significant level α in range [5 − 10]% .

Original: In case thethe residual distribution is significantly different ˜ R d ̸
Revised: In case the residual distribution is significantly different ˜ R d ̸

Original: = R
Revised: = R

Original: i , the alternate hypothesis H 1 : z i causes z j is accepted.
Revised: i , the alternate hypothesis H 1 : z i causes z j is accepted.

Original: D n,m tends to have significantly larger value in alternate case.
Revised: D n,m tends to have significantly larger value in alternate case.

Original: Our method discovers a consistent full causal graph because it is retrieved by testing model invariance over multiple forecast windows.

Original: 5. Experimental data and Results
Revised: 5. Experimental data and Results

Original: We conducted experiments on  synthetically generated time series.
Revised: We conducted experiments on real and synthetically generated time series data.

Original: The synthetic data modelin our experiments has the generic form:
Revised: The synthetic data model has the generic form:

Original: [Equation]
Revised: [Equation]

Original: for j ∈ 1 , ..., N .
Revised: for j ∈ 1 , ..., N .

Original: The correlation coefficients a j are selected from  [0 .
Revised: The correlation coefficients a j are selected from range [0 .

Original: 2 ∼ 1 . 0] .
Revised: 2 − 1 . 0] .

Original: Besides autodependency in  the model, there exist linear and exponential functional dependencies
Revised: Besides autodependency in [Figure] the model, there exist linear and exponential functional dependencies

Original: f i ( z ) .
Revised: f i ( z ) .

Original: We included noise η jt , which represents data from uncorrelated, normally distributed noise with zero mean and variances in the range [0.30 ∼ 0.9].
Revised: We included noise η jt , which represents data from uncorrelated, normally distributed noise with zero mean and variances in the range [0 . 30 − 0 . 9] .

Original: The coupling coefficients c i and lags τ i between variables are set to [0.
Revised: The coupling coefficients c i and lags τ i between variables are set to [0 .

Original: ∼ 1.0] and [0 ∼ 10] respectively.
Revised: 2 − 1 . 0] and [0 − 10] respectively.

Original: The synthetic causal graph contains N=5 variables. To test our method, we generate multiple causal graphs  by changing functional dependencies, causal edges, time delays, dynamic noise and correlation coefficients throughout our experiments.
Revised: To test our method, we generate multiple causal graphs with N = 5 nodes by changing functional dependencies, causal edges, time delays, dynamic noise and correlation coefficients throughout our experiments.

Original: To quantify the performance of our method, we calculate the metrics of accuracy, false positive rate (FPR) and F-score.
Revised: To quantify the performance of applied methods, we calculate the following false positive rate (FPR) and F-score metrics in a variety of generated causal graphs.

Original: [Equation]
Revised: [Equation]

Original: Here, TP is the number of causal links that exist in the  causal graph and also detected by the method; TN is the number of causal links that neither exist in the causal graph nor detected by the method; FP is the number of causal links that do not exist in the causal graph but detected by the method and FN is the number of the causal links that exist in the causal graph but not detected by the method.
Revised: Here, TP is the number of causal links that exist in the ground truth causal graph and also detected by the applied methods; TN is the number of causal links that neither exist in the causal topology nor detected by the causality techniques; FP is the number of causal links that do not exist in the causal graph but detected by the methods and FN is the number of the causal links that exist in the causal graph but not detected by the used methods.

Original: In our experiments we found that methods for generating intervention variables have significant impact on discovering accurate causal graph. To show the effectiveness of proposed knockoff interventions compared to other possible interventions, we tested, using the similar steps described in the previous section, theperformance of following  intervention methods.
Revised: To show the effectiveness of proposed approach where we utilize Knockoffs-based intervention variables, we compare its performance against other widely used causal inference methods. Besides that, to highlight the power of using Knockoffs as intervention variables, we test the proposed approach with the following methods for generating intervention variables.

Original: Distribution mean: The predictor z i is replaced withanother variable z i , that is comprised of the mean of the 
Revised: Distribution mean: The predictor z i is replaced with z i , that is comprised of the mean of the original variable and

Original: original

Original: variable and Gaussian noise.
Revised: Gaussian noise.

Original: z i = 1 r (cid:80) r t =

Revised: z i = 1 r (cid:80) rt =

Original: z i,t + N (0 , Σ) .
Revised: z i,t + N (0 , Σ) .

Original: Uniform distribution: The original variable z i is replaced with another variable  having a uniform distribution which range is defined by the minimum and maximum values of the original variables z i as suggested in (Jiwoong Im et al., 2021)
Revised: Uniform distribution: The original variable z i is replaced by a variable z ′ i having a uniform distribution ranging from minimum to maximum values of the original variables z i as suggested in (Jiwoong Im et al., 2021).

Original: Out-of-distribution: Here, the candidate variable z i that belongs to data distribution
Revised: Out-of-distribution: The variable z i having a distribution

Original: D i is replaced with OOD variable z from another Gaussian distribution D such that D d ̸ = D i (different µ and σ 2 ).
Revised: D i is replaced with another variable sampled from OOD D such that D d ̸ = D i , different µ and σ 2 .

Original: Moreover, z i is selected to beas uncorrelated with the original variable z i as possible.
Revised: Moreover, it is generated to be uncorrelated with the original variables.

Revised: As a real-world application, we conducted experiments on average daily discharges of rivers in the upper Danube basin, provided by the Bavarian Environmental Agency 1 .

Revised: We use measurements from 2017 to 2019 from the Iller at Kempten

Revised: K t , the Danube at Dillingen D t , and the Isar at Lenggries

Original:  Results : We generate multiple causal graphs using synthetic data model and demonstrate the results in terms ofFPR and
Revised: L t . The Iller flows into the Danube within a day, which implies an instantaneous causal link K t → D t and no direct causal links between the pairs K t , L t and D t , L t . Moreover all variables may be confounded by rainfall or other weather conditions (Gerhardus & Runge, 2020), which test the ability of the applied methods to distinguish between spurious associations and true causal relationships. Results : We generate multiple causal graphs using synthetic data model and demonstrate the results in terms of

Original:  F-score for VAR-GC, VARLiNGAM, PCMCI  and DeepAR with all four types of intervention variables represented as 
Revised: FPR and F-score for VAR-GC, VARLiNGAM, PCMCI + and DeepAR with all four types of intervention variables represented as DeepAR-K for Knockoffs, DeepAR-O for

Original: DeepAR-K for knockoffs, DeepAR-O for OOD, DeepARM for mean and DeepAR-U for uniform distribution in Fig 1 and 2.
Revised: OOD, DeepAR-M for mean and DeepAR-U for uniform distribution. We infer causality among nodes of the graph using model invariance through knockoff interventions where we use the KS test (3) to compare the residual distributions of [Figure] [Table]

Revised: the model with and without interventions.

Original: Moreover in Table 1, we present methods accuracy and F-score calculated fromTP, TN, FP and TN in one of our experiments.
Revised: An example is shown in Fig [Table] where we illustrates the causal decision by our method between pair of nodes Z 1 and Z 4 . Based on the ground truth from our generated causal graph, Z 1 causes

Revised: Z 4 while the reverse in not true.

Revised: Our method accurately identifies Z 1 → Z 4 , a true positive and Z 4 ↛ Z 1 , a true negative.

Original: Overall DeepAR-K performs better than others in accurate causal discovery.
Revised: Overall DeepAR-K performs better than others in accurate causal discovery.

Original: Since knockoffs has strong ability to control false discoveries as the framework was specially developed for, we can see low FPR for knockoff interventions as shown in Fig 1 compared to other methods and intervention types.
Revised: Since the Knockoffs-framework is originally developed to control false discoveries, we can see low FPR for Knockoffs-based interventions as shown in Fig 2 (a) compared to other methods and intervention types.

Original: At the same time thedetection power of DeepAR-K to identify true causal links is high as illustrated in Fig 2  which could be because of uncorrelated property of the knockoffs with their originals.
Revised: At the same time, the power of DeepAR-K to identify true causal links is high as illustrated in Fig 2 (b), which could be attributed to the uncorrelated property of the Knockoffs with their originals.

Original: The closest to our proposed DeepAR-K in terms of achieving high F-score and low FPR are DeepAR-O and DeepARU.
Revised: The closest to our proposed DeepAR-K in terms of achieving high F-score and low FPR is DeepAR-U.

Original: Since we have non-linear links in our generated causal graphs, methods like VAR-GC, VARLiNGAM and PCMCI struggle to estimate accurate causality among nodes of  the graph.
Revised: PCMCI + also achieves relatively better FPR but shows low power for true causal detection along with VAR-GC and VARLiNGAM. The low detection power of these methods might be because of nonlinearity in the generated causal graphs. For river discharges data, we have one contemporaneous link

Original: VAR-GC suffered the most and obtained a high FPR as it assumes linearity among variables of time series.
Revised: K t → D t as per description in the dataset, and we expect the applied methods to identify it. Our proposed method along with all other methods except DeepAR-Mean accurately detected K t → D t as shown in Table 1.

Original: [Table]
Revised: Moreover, our method does not report any false causal link. However, the rest of the methods show spurious associations between various pairs of river discharges time series. DeepAR-OOD incorrectly detected the reversed causal link D t → K t too. VAR-GC and DeepAR-M identified a bidirectional link between D t and L t which may be because of the weather acting as a confounder since there is no expected direct causal relationship.

Revised: DeepAR-U, DeepAR-OOD and PCMCI

Revised: + have one false link detection each in different pairs.

Original: 6. Conclusion
Revised: 6. Conclusion

Original: We proposed to test model invariance across interventional environments over multiple forecast windows to discover full causal graph.
Revised: We proposed to test model invariance across interventional environments over multiple forecast windows to discover full causal graph in multivariate nonlinear systems.

Original: DeepAR is used to learn non-linear relationship in multivariate time series.
Revised: DeepAR is used to learn nonlinear relationship in multivariate time series.

Original: The model invariance is tested using knockoff interventions.
Revised: The model invariance is tested via Knockoffs-based interventions for causal inference.

Original: We used synthetic  time series data to evaluate our method.
Revised: We used synthetic and real time series data to evaluate applied methods.

Original: The results confirm that the estimation of the environments using Knockoffs to test model invariance for causal inference yield better results in comparison to others.
Revised: The results indicate that the estimation of the environments using Knockoff framework to test model invariance for causal inference yield better results in comparison to other intervention types.

Original: Results also indicate that the proposed method outperforms VAR-GC, VARLiNGAM and PCMCI, however having a high computation time.
Revised: Results also show that the proposed method outperforms VAR-GC, VARLiNGAM and PCMCI + , however having a higher computation time.

Original: Further we will test our method on  real-world time series and extend the work for non-stationary data.
Revised: Further we aim to extend the current work for non-stationary settings and test our method on other real-world time series.

Revised: Acknowledgements

Revised: This work is funded by the Carl Zeiss Foundation within the scope of the program line “Breakthroughs: Exploring

Revised: Intelligent Systems” for “Digitization — explore the basics, use applications” and the DFG grant SH 1682/1-1.