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 nonlinear multivariate systems 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 interactions in time series using DeepAR and then expose the model to different environments using Knockoffs-based interventions 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: Knockoff samples are pairwise exchangeable, in-distribution and statistically 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 predictors.
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 evaluate our method on real and synthetically generated time series.
Revised: We evaluate our method on real and synthetically generated time series.

Original: Overall our method outperforms other widely used causality methods, i.e, VAR Granger causality, VARLiNGAM and PCMCI + .
Revised: Overall our method outperforms other widely used causality methods, i.e, VAR Granger causality, VARLiNGAM and PCMCI + .

Revised: The code and data can be found at: https://github.com/wasimahmadpk/deepCausality

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.
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 Knockoffs (Barber & Cand`es, 2015; Candes et al., 2017) based 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; HeinzeDeml 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 at the 2022 Workshop on Spurious Stability conditional distribution of the response given its causal predictors remains invariant for all observations over different settings.

Original: 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.
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, which is based on recurrent neural networks (RNNs) and generate probabilistic forecast.
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: 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.
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 either omit or replace model predictors with a different representation.
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 cannot handle missing input or out-of-distribution (OOD) data, we propose to use the theoretically well-established Knockoffs framework 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: Knockoffs 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.

Original: Moreover,
Revised: Moreover,

Original: 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.
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: We highlight the effectiveness of Knockoffs as interventional variables by comparing it with other intervention methods such as OOD, mean and 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: 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
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: (Hyv¨arinen et al., 2010), PCMCI + (Runge, 2020) as well as other applied intervention methods.
Revised: (Hyv¨arinen et al., 2010), PCMCI + (Runge, 2020) as well as other applied intervention methods.

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: ICP 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 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.
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 for causal inference in nonlinear systems.
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: Previously we used deep networks with Knockoffs-based interventions to estimate Granger causality in nonlinear multivariate time series (Ahmad et al., 2021).
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) utilize 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: The widely used PCMCI (Runge et al., 2019) method applies 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 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).
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-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.
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 and the Knockoffs framework.
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 (Salinas et al., 2020).
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 RNNs (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 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.
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, lesser hand-crafted feature engineering is needed in
Revised: Moreover, lesser hand-crafted feature engineering is needed in

Original: DeepAR to capture complex, group-dependent behavior.
Revised: DeepAR to capture complex, group-dependent behavior.

Original: 3.2. Knockoffs
Revised: 3.2. Knockoffs

Original: The Knockoff 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 knockoff copies 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 ) , represent interventional environments while maintaining the in-distribution and decorrelation property of the observational environment.
Revised: , (cid:101) Z n ) , represent interventional environments while maintaining the in-distribution and decorrelation property of the observational environment.

Original: Knockoff samples are statistically null-variables, generated without knowing model response.
Revised: Knockoff samples are statistically null-variables, generated without knowing model response.

Original: Moreover, knockoffs 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]

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

Original: P Z ( z ) P (cid:101)
Revised: P Z ( z ) P (cid:101)

Original: Z | Z ( (cid:101)
Revised: Z | Z ( (cid:101)

Original: z | z ) , is pairwise ( z j , (cid:101)
Revised: 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 knock- offs (cid:101)

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

Original: P (cid:101) Z | Z ( .
Revised: P (cid:101) Z | Z ( .

Original: | Z i, ∗ ) =
Revised: | Z i, ∗ ) =

Original: N n (( I n − S Σ − 1 )
Revised: N n (( I n − S Σ − 1 )

Original: Z i, ∗ , 2 S − S Σ − 1 S ) (2) 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 to 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 2 n (cid:18) 

Revised: N 2 n (cid:18)

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

Original: Here S is to be maximized which represents the difference between the original variables and their knockoff version.
Revised: Here S is to be maximized which represents the difference between the original variables and their knockoff version.

Original: 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.
Revised: 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 relations 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: An extension of the current work to deal with non-stationary settings is in progress.
Revised: An extension of the current work to deal with non-stationary settings is in progress.

Original: 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.
Revised: 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: 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.
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 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.
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 variables 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: To generate knockoff variables, 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 use mixture of Gaussian models as implemented in (Gimenez et al., 2019) to obtain knockoff samples.
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 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.
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: Given observed variable Z ∼ N ( µ, Σ) , we aim to build its knockoff copy (cid:101)
Revised: Given observed variable Z ∼ N ( µ, Σ) , we aim to build its knockoff copy (cid:101)

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

Original: ∗ , ∗∗ ) 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 original and knockoff version.
Revised: We have the same µ for both original and knockoff version.

Original: The
Revised: The

Original: 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.
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.

Original: 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 .
Revised: 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 relationships.
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 using 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 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 , . . .
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 distributions using Kolmogorov–Smirnov (KS) test (Smirnov, 1939) for causal inference.
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 in range [5 − 10]% .
Revised: ˜ R d ≈ R i with a significant level α in range [5 − 10]% .

Original: In case the 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: 5. Experimental data and Results
Revised: 5. Experimental data and Results

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

Original: The synthetic data model has the  generic form:
Revised: The synthetic data model has the [Figure] 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 range [0 .
Revised: The correlation coefficients a j are selected from range [0 .

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

Original: Besides autodependency in[Figure] the model, there exist linear and exponential functional dependencies
Revised: Besides autodependency in 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: 2 − 1 . 0] and [0 − 10] respectively.
Revised: 2 − 1 . 0] and [0 − 10] respectively.

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

Original: [Equation]

Original: 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: 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.
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.

Original: Besides that, to highlight the power of using
Revised: Besides that, to highlight the power of using

Original: Knockoffs as intervention variables, we test the proposed approach with the following methods for generating intervention variables.
Revised: 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 with z i , that is comprised of the mean of the original variable and Gaussian noise.
Revised: Distribution mean: The predictor z i is replaced with z i , that is comprised of the mean of the original variable and Gaussian noise.

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

Original: =

Revised: =

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

Original: Uniform distribution: The original variable z i is replaced by a variable z ′
Revised: Uniform distribution: The original variable z i is replaced by a variable z ′

Original: 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).
Revised: 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: The variable z i having a distribution
Revised: Out-of-distribution: The variable z i having a distribution

Original: D i is replaced with another variable sampled from OOD
Revised: D i is replaced with another variable sampled from OOD

Original: D such that D d ̸ = D i , different µ and σ 2 .
Revised: D such that D d ̸ = D i , different µ and σ 2 .

Original: Moreover, it is generated to be uncorrelated with the original variables.
Revised: Moreover, it is generated to be uncorrelated with the original variables.

Original: 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: 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 .

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

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

Original: L t .
Revised: L t .

Original: 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 .
Revised: 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 .

Original: 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.
Revised: 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.

Original: Results : We generate multiple causal graphs using synthetic data model and demonstrate the results in terms of
Revised: Results : We generate multiple causal graphs using synthetic data model and demonstrate the results in terms of

Original: 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
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

Original: for
Revised: for

Original: OOD, DeepAR-M for mean and DeepAR-U for uniform distribution.
Revised: OOD, DeepAR-M for mean and DeepAR-U for uniform distribution.

Original: 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] the model  with and without interventions.
Revised: We infer causality among nodes of the graph using model invariance through knockoff interventions where we use the KS test (3) to compare the distribution of model residuals with and without interventions.

Original: 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 .
Revised: In Fig 1, we illustrate the causal decision by our method between pair of nodes Z 1 and Z 4 .

Original: Based on the ground truth from our generated causal graph, Z 1 causes Z 4 while the reverse in not true.
Revised: Based on the ground truth from our generated causal graph, Z 1 causes Z 4 while the reverse in not true.

Original: Our method accurately identifies Z 1 → Z 4 , a true positive and Z 4 ↛ Z 1 , a true negative.
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 the Knockoffs-framework is originally developed to control false discoveries, we can see low FPR for Knockoffs-based interventions as shown in
Revised: Since the Knockoffs-framework is originally developed to control false discoveries, we can see low FPR for Knockoffs-based [Figure] [Table]

Original:  Fig 2 (a) compared to other methods and intervention types.
Revised: interventions as shown in Fig 2 (a) compared to other methods and intervention types.

Original: 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.
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 is DeepAR-U. PCMCI + also achieves relatively better FPR but shows low power for true causal detection along with VAR-GC and VARLiNGAM.
Revised: The closest to our proposed DeepAR-K in terms of achieving high Fscore and low FPR is DeepAR-U. PCMCI + also achieves relatively better FPR but shows low power for true causal detection along with VAR-GC and VARLiNGAM.

Original: The low detection power of these methods might be because of nonlinearity in the generated causal graphs.
Revised: The low detection power of these methods might be because of nonlinearity in the generated causal graphs.

Original: For river discharges data, we have one contemporaneous link
Revised: For river discharges data, we have one contemporaneous link

Original: K t → D t as per description in the dataset, and we expect the applied methods to identify it.
Revised: K t → D t as per description in the dataset, and we expect the applied methods to identify it.

Original: Our proposed method along with all other methods except DeepAR-Mean accurately detected K t → D t as shown in Table 1.
Revised: Our proposed method along with all other methods except DeepAR-Mean accurately detected K t → D t as shown in Table 1.

Original: Moreover, our method does not report any false causal link.
Revised: Moreover, our method does not report any false causal link.

Original: However, the rest of the methods show spurious associations between various pairs of river discharges time series.
Revised: However, the rest of the methods show spurious associations between various pairs of river discharges time series.

Original: DeepAR-OOD incorrectly detected the reversed causal link D t → K t too.
Revised: DeepAR-OOD incorrectly detected the reversed causal link D t → K t too.

Original: 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: 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.

Original: DeepAR-U, DeepAR-OOD and PCMCI
Revised: DeepAR-U, DeepAR-OOD and PCMCI

Original: + have one false link detection each in different pairs.
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 in multivariate nonlinear systems.
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 nonlinear relationship in multivariate time series.
Revised: DeepAR is used to learn nonlinear relationship in multivariate time series.

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

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

Original: The results indicate that the estimation of the environments using
Revised: The results indicate that the estimation of the environments using

Original: Knockoff framework to test model invariance for causal inference yield better results in comparison to other intervention types.
Revised: Knockoff framework to test model invariance for causal inference yield better results in comparison to other intervention types.

Original: Results also show that the proposed method outperforms VAR-GC, VARLiNGAM and PCMCI + , however having a higher 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 aim to extend the current work for non-stationary settings and test our method on other real-world time series.
Revised: Further we aim to extend the current work for non-stationary settings and test our method on other real-world time series.

Original: Acknowledgements
Revised: Acknowledgements

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

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