Title: Directed Cyclic Graph for Causal Discovery from Multivariate Functional Data
Venue: NeurIPS 2023
Accepted: yes
Keywords: Causal Embedding

Causal Discovery

Multivariate Functional Data

Directed Cyclic Graph

Causal Structure Learning

Bayesian Inference
Abstract: Discovering causal relationship using multivariate functional data has received a significant amount of attention very recently. In this article, we introduce a functional linear structural equation model for causal structure learning when the underlying graph involving the multivariate functions may have cycles. To enhance interpretability, our model involves a low-dimensional causal embedded space such that all the relevant causal information in the multivariate functional data is preserved in this lower-dimensional subspace. We prove that the proposed model is causally identifiable under standard assumptions that are often made in the causal discovery literature. To carry out inference of our model, we develop a fully Bayesian framework with suitable prior specifications and uncertainty quantification through posterior summaries. We illustrate the superior performance of our method over existing methods in terms of causal graph estimation through extensive simulation studies. We also demonstrate the proposed method using a brain EEG dataset.

Directed Cyclic Graph for Causal Discovery from Multivariate Functional Data

Abstract

Discovering causal relationship using multivariate functional data has received a significant amount of attention very recently. In this article, we introduce a functional linear structural equation model for causal structure learning when the underlying graph involving the multivariate functions may have cycles. To enhance interpretability, our model involves a low-dimensional causal embedded space such that all the relevant causal information in the multivariate functional data is preserved in this lower-dimensional subspace. We prove that the proposed model is causally identifiable under standard assumptions that are often made in the causal discovery literature. To carry out inference of our model, we develop a fully Bayesian framework with suitable prior specifications and uncertainty quantification through posterior summaries. We illustrate the superior performance of our method over existing methods in terms of causal graph estimation through extensive simulation studies. We also demonstrate the proposed method using a brain EEG dataset.

1 Introduction

Motivation. Multivariate functional data arise in many fields such as biomedical research [Wei and Li, 2008, Chiou and Müller, 2016], environmental science [Korte-Stapff et al., 2022], finance [Kowal et al., 2017], plant science [Wong et al., 2019, Park et al., 2022], and sport science [V olkmann et al., 2021] where multiple variables are measured over time or other domains. The increasing availability of functional data in these fields provides us with great opportunities to discover causal relationships among random functions for the better understanding of complex systems, which is helpful for various machine learning and statistics tasks such as representation learning [Schölkopf et al., 2021], fairness [Tang et al., 2023], transfer learning [Rojas-Carulla et al., 2018], and reinforcement learning [Zeng et al., 2023]. One motivating example is electroencephalography (EEG) where electrical activity from the brain is recorded non-invasively from electrode channels by placing them on the scalp or directly on the surface of the brain. Given its continuous nature and the short time separation between the adjacent measuring points, it is natural to treat the data at each brain location/region as a function over time. A relevant scientific goal is to estimate brain effective connectivity among different regions, which will potentially allow us to make better decisions, design more effective interventions, and avoid unintended consequences. However, existing structural equation model (SEM) based causal discovery methods assume acyclic relationships among the random functions by imposing a directed acyclic graph (DAG) structure, which may be too restrictive for many real applications. For example, there are strong indications that in brain effective connectivity studies, due to reciprocal polysynaptic connections, the brain regions are far from exhibiting acyclicity [Friston, 2011, Markov et al., 2012], and that in genetic pathways, due to the presence of multiple upstream regulators and downstream targets for every signaling component, feedback loops/directed cycles are regular motifs [Brandman

and Meyer, 2008]. Thus, in light of the prevalence of cycles in complex systems, it is desirable to have a flexible model for causal discovery among random functions that can account for such cyclic causal structures.

Challenges. Causal discovery for multivariate functional data in the presence of cycles is an inherently difficult problem that is not yet well understood. We highlight three prominent challenges. (i) Functional data are infinite-dimensional in nature. It may so happen that the low-frequency spectrum of one curve might causally influence the high-frequency spectrum of another curve. This demands identification of pertinent features that can be used to create a finite-dimensional representation of the data, which is easier to work with and analyze. However, the challenge is that we may not know a priori what the relevant features are when dealing with infinite-dimensional objects. Blind adoption of standard (non-causal-adaptive) low-dimensional features can lead to errors or inaccuracies. (ii) Although the identifiability of causal models for multivariate functional data in the absence of cycles has been established in recent works [Zhou et al., 2022b, Lee and Li, 2022], showing identifiability of causal models from multivariate data, let alone multivariate functions, is still a challenging and complex task in cases where causal relationships are obscured by the presence of cycles. (iii) It is common that functional data are only observed over discrete time points with additional noises. Such incomplete and noisy observations of the functions add another layer of difficulty in probing the causal relationships of interest.

Related work. Causal discovery from multivariate functional data has been studied by a few recent works [Zhou et al., 2022b, Lee and Li, 2022, Yang and Suzuki, 2022], which have already shown some promising results in discovering causality in, e.g., EEG data and longitudinal medical record data. However, all of them are limited to DAGs, which do not allow inference of cyclic causality. While there has been a surge of research on causal discovery methods for scalar random variables in the presence of feedback loops/cycles over the last few decades [Richardson, 1996, Lacerda et al., 2008, Mooij et al., 2011, Hyttinen et al., 2012, Huang et al., 2019, Mooij and Heskes, 2013, Mooij and Claassen, 2020, Zhou et al., 2022a], none of these approaches have been extended to discovering causal dependencies among random functions in multivariate settings. Therefore, how to handle cyclic causal relationships among multivariate functional data while addressing the aforementioned challenges remains a largely unsolved problem.

Contributions. In this paper, we propose an operator-based non-recursive linear structural equation based novel causal discovery framework that identifies causal relationships among functional objects in the presence of cycles and additional measurement/sampling noises. Our major contribution is four-fold.

We consider a causal embedding of the functional nodes into a lower-dimensional space for dimension reduction that adapts to causal relationships.

We prove that the causal graph of the proposed model is uniquely identifiable under standard causal assumptions.

We capture within-function dependencies using a data-driven selection of orthonormal basis that is both interpretable and computationally efficient.

To perform inference and uncertainty quantification from finite-sample data, we adopt a fully Bayesian hierarchical formulation with carefully selected prior distributions. Posterior inference is performed using Markov chain Monte Carlo (MCMC). We demonstrate the effectiveness of the proposed method in identifying causal structure and key parameters through simulation studies and apply the framework to the analysis of brain EEG data, illustrating its real-world applicability. Code will be made available on the project's website on Github.

2 Model definition and causal identifiability

2.1 Notations

Let [ p ] = { 1 , . . . , p } for any positive integer p . A causal directed cyclic graph (DCG) is a graph G = ( V , E ) , which consists of a set of vertices or nodes V = [ p ] representing a set of random variables

and a set of directed edges E = { ℓ → j | j, ℓ ∈ V} representing the direct causal relationships among the random variables. In a DCG, we do not assume the graph to be acyclic. A causal DCG model is an ordered pair ( G , P ) where P is a joint probability distribution over V (more rigorously, the random variables that V represents) that satisfies conditional independence relationships encoded by the causal DCG G . A simple directed cycle is a sequence of distinct vertices { v 1 , . . . , v k } such that the induced subgraph by these vertices is v 1 →··· → v k → v 1 . For a vertex j ∈ V , we use pa( j ) to denote the set of parents (direct causes).

2.2 Model framework

Consider a multivariate stochastic process Y = ( Y 1 , . . . , Y p ) ⊤ where each Y j is defined on a compact domain T j ⊂ R . Without loss of generality, we assume T 1 = · · · = T p = [0 , 1] . Suppose Y j ∈ H j where H j is a Hilbert space of functions defined on T j . We let ⟨· , ·⟩ denote the inner product of H j . We propose a causal model that captures the relationships among Y 1 , . . . , Y p .

Our proposed model considers an operator-based non-recursive linear structural equation model on the random functions Y as

where B jℓ is a linear operator that maps H ℓ to H j , and f j ∈ H j is an exogenous stochastic process. Clearly, for any j, ℓ ∈ V such that the edge ℓ → j ∈ E , B jℓ is not a null operator. Now by stacking the p equations in (1), we obtain

where B = ( B jℓ ) p j,ℓ =1 is a matrix of operators and f = ( f 1 , . . . , f p ) ⊤ is a p -variate stochastic process. In DAGs, the causal effect matrix can be arranged into a lower block triangular structure given a topological/causal ordering. But since our model allows for cycles, we have no such restriction on the structure of the operator matrix B except that B jj , ∀ j ∈ [ p ] , is null, i.e., no self-loops.

Model (1) is infinite-dimensional and hence challenging to estimate and interpret. To alleviate such difficulties, we consider a low-dimensional causal embedding structure. Specifically, we assume that the causal relationships are preserved in an unknown low-dimensional subspace D j of H j . Denote the dimension of D j by K j . Let P j and Q j be the projection onto D j and its orthogonal complement in H j respectively. We assume B jℓ = P j B jℓ P ℓ , which implies that causal effects can be fully described within the low-dimensional subspaces {D j } p j =1 . As such, (1) can be split into

We assume that P j f j and Q j f j are independent of each other. Now, by defining α j = P j Y j and ϵ j = P j f j , ∀ j ∈ [ p ] , (3) can be compactly written as

In practice, the random functions in Y can only be observed over a finite number of (input) locations, possibly with measurement errors. More specifically, for each random function Y j , we observe { ( t ju , X ju ) } m j u =1 , where X ju ∈ R is the measurement of Y j at location t ju ∈ T j and m j is the number of measurements obtained from Y j . Defining β j = Q j Y j , we consider the following measurement model:

with independent noises e ju ∼ N (0 , σ j ) , ∀ u ∈ [ m j ] .

More compactly, (5) can be written as

where X = ( X ⊤ 1 , . . . , X ⊤ p ) ⊤ , α ( t ) = ( α 1 ( t 1 ) ⊤ , . . . , α p ( t p ) ⊤ ) ⊤ , β ( t ) = ( β 1 ( t 1 ) ⊤ , . . . , β p ( t p ) ⊤ ) ⊤ and e = ( e ⊤ 1 , . . . , e ⊤ p ) ⊤ with X j = ( X j 1 , . . . , X jm j ) ⊤ , α j ( t j ) = ( α j ( t j 1 ) , . . . , α j ( t jm j )) ⊤ , β j ( t j ) = ( β j ( t j 1 ) , . . . , β j ( t jm j )) ⊤ and e j = ( e j 1 , . . . , e jm j ) ⊤ .

We call our proposed model, FENCE , which stands for ' F unctional E mbedded N odes for C yclic causal E xploration', reflecting its purpose.

2.3 Causal identifiability

In this section, we shall show that the graph structure of the proposed FENCE model is identifiable for functional data measured discretely with random noises under several causal assumptions. We start by defining causal identifiability and state our assumptions.

̸

Definition 2.1. ( Causal Identifiability ) Suppose Y is a p -variate random function and X is the observed noisy version of Y given by (6). Assume X follows FENCE model S = ( G , P ) where G is the underlying graph and P is the joint distribution of X over G . We say that S is causally identifiable from X if there does not exist any other S ∗ = ( G ∗ , P ∗ ) with G ∗ = G such that the joint distribution P ∗ on X induced by G ∗ is equivalent to P induced by G .

In other words, for a causal graph to be identifiable, there must not exist any other graph such that the joint distributions induced by the two different graphs are equivalent. Next, we list and discuss a few assumptions to establish the causal identifiability of the proposed model.

Assumption 1. (Causal Sufficiency) The model S = ( G , P ) is causally sufficient, i.e., there are no unmeasured confounders.

Assuming no unmeasured confounders keeps the causal discovery task more manageable especially for cyclic graphs with purely observational data.

Assumption 2. (Disjoint Cycles) The cycles in G are disjoint, i.e., no two cycles in the graph have two nodes that are common to both.

Assuming disjoint cycles induces a natural topological ordering and forms a directed acyclic hypergraph-like structure within the DCG. The same assumption was made in Lacerda et al. [2008].

Assumption 3. (Stability) For the model S , the moduli of the eigenvalues of the finite rank operator B are less than or equal to 1 , and none of the real eigenvalues are equal to 1 .

According to Fisher, 1970, the SEM in (4) can be viewed as being in a state of equilibrium, where the finite rank operator B represents coefficients in a set of dynamical equations that describe a deterministic dynamical system observed over small time intervals as the time lag approaches zero. The eigenvalue conditions are deemed necessary and sufficient for the limiting behavior to hold, as argued by Fisher, 1970. Such an assumption is widely adopted in e.g., econometrics, and Lacerda et al., 2008 made this assumption as well.

Assumption 4. (Non-Gaussianity) The exogenous variables have independent mixture of Gaussian distributions. i.e., ϵ jk ind ∼ ∑ M jk m =1 π jkm N ( µ jkm , τ jkm ) with M jk ≥ 2 .

The assumption of non-Gaussianity on the exogenous variables has been proven useful in causal discovery as it induces model identifiability in the linear SEM framework [Aapo and Petteri, 1999, Shimizu et al., 2006, Lacerda et al., 2008, Spirtes and Zhang, 2016]. Mixture of Gaussian can approximate any continuous distribution arbitrarily well given a sufficiently large number of mixture components [Titterington et al., 1985, McLachlan and Peel, 2000, Rossi, 2014]. It is also easy to sample, which facilitates our posterior inference.

Assumption 5. (Non-causal dependency) We assume β ( t ) = C ( t ) γ , where C ( t ) = diag ( C 11 ( t 1 ) , . . . , C pp ( t p )) and γ represent another exogenous component of the model. Here C jj ( t j ) is a mixing matrix that mixes the independent entires in γ to generate temporal dependence within the j -th block. We assume γ jk ind ∼ ∑ M jk m =1 π ′ jkm N ( µ ′ jkm , τ ′ jkm ) with M jk ≥ 1 .

̸

Since the model assumes that all causal information in Y is preserved in the lower-dimensional space D j and not in its orthogonal complement, it is apparent that while each β j ( t j ) within a block can have temporal dependence, it is independent of β ℓ ( t ℓ ) when j = ℓ and j, ℓ ∈ [ p ] .

For some basis { ϕ jk } K j k =1 that spans the low-dimensional causal embedded space D j , α j in (5) can be further expanded by, α j ( t ju ) = ∑ K j k =1 ˜ α jk ϕ jk ( t ju ) . Therefore (6) can be written more compactly as

Assumption 6. (Sufficient sampling locations) The basis matrix Φ ( t ) of size ∑ p j =1 m j × ∑ p j =1 K j has a full column rank.

This assumption implies enough sampling locations, over which each random function Y j is observed, to capture all the causal information that Y j contains.

Given these six assumptions, our main theorem establishes the causal identifiability of the proposed model.

Theorem 2.1. Under Assumptions 1-6, S = ( G , P ) is causally identifiable.

The proof essentially involves two steps as shown in Figure 1. On the left-hand side (LHS) of the diagram, we depict the hypergraph-like structure that emerges when assuming the existence of disjoint cycles (Assumption 2), whereas, on the right-hand side (RHS), we offer a magnified view of the hypernodes (nodes containing simple directed cycle). Our approach to proving causal identifiability progresses from the LHS to the RHS. That is, we first prove the identifiability of the hypergraph-like structure depicted on the LHS of Figure 1, and then we proceed to establish the identifiability of each simple directed cycle within every hypernode in the hypergraph. The detailed exposition of the proof can be found in Section A of the Supplementary Materials.

O

QO

3 Bayesian model formulation

In this section, we will describe the inference procedure of the proposed model. A straightforward approach would be a two-step procedure where the first step performs functional principal component analysis on each function marginally to reduce the dimension, and then the second step learns causal structure based on the principal components. However, this simple approach has several disadvantages. First, the estimated functional principal components that explain the most variation of each individual function marginally may not optimally capture the cause-effect dependence relationships among different functions. Second, this procedure is unreliable since estimation uncertainty fails to propagate correctly from the first step to the second step. As such, we propose a fully Bayesian approach, which reduces the dimension of functional data adaptively for causal structure learning.

3.1 Model parameters

Let E = ( E jℓ ) p j,ℓ =1 denote the adjacency matrix where E jℓ = 1 indicates the existence of a directed edge from node ℓ to node j , and 0 otherwise. Let { ϕ k } S k =1 be a set of S common unknown basis

functions that approximate each random function Y j i.e., Y j = ∑ S k =1 ˜ α jk ϕ k where { ˜ α jk } S k =1 denote the set of basis coefficients. Note that { ϕ k } is not the basis for the lower-dimensional causal embedded subspace D j . However, we assume that the first K j of them actually spans D j and our goal is to hunt for them through a properly designed inference procedure. Moreover, according to our assumption, we build our SEM on the first K j of the basis coefficients ˜ α j = (˜ α j 1 , · · · , ˜ α jK j ) ⊤ .

Defining

¯

α

j

= (˜

α

j,K

+1

,

· · ·

,

˜

α

)

⊤

with

¯

α

j

=

γ

j

, jointly they can be written as

(( B jℓ ( a, b )) K j ,K l a =1 ,b =1 ) p j,ℓ =1 with B jj = 0 since we assume the absence of self loops. To carry out inference, we assume ˜ ϵ jk , γ jk ind ∼ ∑ M jk m =1 π jkm N ( µ jkm , τ jkm ) .

3.2 Adaptive basis expansion

As the ϕ k 's are specifically useful for restricting the original function space for each Y j to a lowerdimensional causally embedded smooth space of dimension K j , we make the basis { ϕ k } adaptive for causal structure learning by further expanding them with known spline basis functions [Kowal et al., 2017], ϕ k ( · ) = ∑ R r =1 A kr b r ( · ) , where b = ( b 1 , . . . , b R ) ⊤ is the set of fixed cubic B-spline basis functions with equally spaced knots and A k = ( A k 1 , . . . , A kR ) ⊤ are the corresponding spline coefficients. Since we do not fix A k 's a priori , the basis functions ϕ k 's can be learned from data a posteriori and hence are adaptive to both data and causal structure (i.e., the basis functions, the functional data, and the causal graph are dependent in their joint distribution).

3.3 Prior specifications

Prior on spline coefficients. The prior on A k is chosen to serve multiple purposes. (i) It sorts the basis functions by decreasing smoothness and therefore helps to identify the spanning set of size K j for the underlying smooth causally embedded space D j . (ii) Although not a strict requirement for modelling purpose, it forces ϕ k 's to be orthonormal, i.e. ∫ ϕ k ( ω ) ϕ k ′ ( ω ) dω = I ( k = k ′ ) . As such, the orthogonality constraints help eliminate any information overlap between the basis functions, which keeps the total number of necessary basis functions that actually contribute to the causal structure learning to a minimum. (iii) It regularizes the roughness of ϕ k 's to prevent overfitting.

For (iii), more specifically, we restrict the roughness of the basis functions ϕ k ( · ) by assigning a prior that penalizes its second derivatives [Gu, 1992, Wahba, 1978, Berry et al., 2002]:

where Ω -is the pseudoinverse of Ω = ∫ b ′′ ( t )[ b ′′ ( t )] ⊤ dt . Let Ω = UDU ⊤ be the singular value decomposition of Ω . Following Wand and Ormerod, 2010, to facilitate computation, we reparameterize ϕ k ( · ) = R ∑ r =1 ˜ A kr ˜ b k ( · ) with ˜ b ( · ) = (1 , t, b T ( · ) ˜ U ˜ D -1 2 ) ⊤ where ˜ D is the ( R -2) × ( R -2) submatrix of D corresponding to non-zero singular values (note that the rank of Ω is R -2 by definition) and ˜ U is the corresponding R × ( R -2) submatrix of U . This induces a prior on ˜ A k given by

In other words, the intercept and the linear term are unpenalized but the non-linear terms are penalized, the degree of which is controlled by λ k . In practice, we set the first two diagonal elements of S k as 10 8 . We constrain the regularization parameters λ 1 > · · · > λ S > 0 by putting a uniform prior:

which implies that the smoothness of ϕ k ( · ) decreases as k gets larger.

j

Priors on the adjacency matrix. We propose to use an independent uniform-Bernoulli prior on each entry E jℓ of E , i.e., E jℓ | ρ ind ∼ Bernoulli ( ρ ) and ρ ∼ Uniform (0 , 1) . The marginal distribution of E with ρ integrated out is given by

̸

̸

Now, for example, if E 0 denotes the null adjacency matrix and E 1 denotes the adjacency matrix with only one edge, then we can see that p ( E 0 ) /p ( E 1 ) = p 2 -p . Therefore, an empty graph is favored over a graph with one edge by a factor of p 2 -p , and, importantly, this penalty increases with p . Thus, the uniform-Bernoulli prior prevents false discoveries and leads to a sparse network by increasing the penalty against additional edges as the dimension p grows.

Prior on the causal effect matrix. Now given E , we assume an independent spike and slab prior on the entries of B = ( B jℓ ) p j,ℓ =1 :

where MVN ( B jℓ ; 0 , γ I K j , I K ℓ ) is a matrix-variate normal distribution with row and column covariance matrices as γ I K j and I K ℓ , respectively. We assume a conjugate inverse-gamma prior on the causal effect size, γ ∼ InverseGamma ( a γ , b γ ) . We choose a γ = b γ = 1 . We fix s = 0 . 02 so that when E jℓ = 0 , B jℓ is negligibly small.

Priors on the parameters of the Gaussian mixture distribution. We choose conjugate priors for the parameters of the Gaussian mixture distribution:

We have fixed values for the hyperparameters, β = 1 , a µ = 0 , b µ = 100 , a τ = b τ = 1 .

Prior on the noise variances. We assume a conjugate prior for σ j ∼ InverseGamma ( a σ , b σ ) , ∀ j ∈ [ p ] . We choose a σ = b σ = 0 . 01 .

We simulate posterior samples through Markov chain Monte Carlo (MCMC). Details are given in Section B of the Supplementary Materials. Sensitivity analyses will be conducted to test the hyperparameters including ( a γ , b γ ) , ( a τ , b τ ) , ( a σ , b σ ) , s, R, S, M and β .

4 Simulation study

Data generation The data were simulated according to various combinations of sample size ( n ), number of nodes ( p ), and grid size ( m j = d ∀ j ∈ [ p ] ) where n ∈ { 75 , 150 , 300 } , p ∈ { 20 , 40 , 60 } , and d ∈ { 125 , 250 } . The grid evenly spans the unit interval [0 , 1] ; the results with unevenly spaced grids are presented in Section C of the Supplementary Materials. The true causal graph G was generated randomly with edge formation probability 2 /p . Given G , each non-zero block B jℓ of the causal effect matrix was generated from the standard matrix-variate normal distribution. We set the true number of basis functions to be K = 4 . In order to generate K = 4 orthonormal basis functions, we first simulated unnormalized basis functions by expanding them further with 6 cubic B-spline basis functions where the coefficients were drawn from the standard normal distribution and then empirically orthonormalized them. The basis coefficients ˜ α were generated following (8) with the exogenous variables ˜ ϵ j drawn independently from Laplace distribution with location parameter µ = 0 and scale parameter b = 0 . 2 . We have also considered other non-Gaussian distributions for the exogenous variables; the corresponding results are provided in Section C of the Supplementary Materials. Finally, noisy observations were simulated following (6) with the signal-to-noise ratio, i.e., the mean value of | Y ( i ) j ( t ) | /σ j across all i and t , set to 5. Here, superscript ( i ) denotes the i th sample, where i ∈ [ n ] .

For the implementation of the proposed FENCE, we fixed the number of mixture components to be 10 and ran MCMC for 5,000 iterations (discarding the first 2,000 iterations as burn-in and retaining every 5th iteration after burn-in). The causal graph G was then estimated by using the median probability model [Barbieri and Berger, 2004], i.e., by thresholding the posterior probability of inclusion at 0.5.

Methods for comparison. We compared our method with fLiNG [Zhou et al., 2022b], a recently proposed directed acyclic graph (DAG) for multivariate functional data. Codes for Lee and Li [2022], Yang and Suzuki [2022] are not publicly available. Hence for more comparison, we considered two ad hoc two-step approaches. In the first step of both approaches, we obtained the basis coefficients by carrying out functional principal component analysis (fPCA) using the fdapace [Zhou et al., 2022c] package in R. Then in the second step, given the basis coefficients, we estimated causal graphs using existing causal discovery methods, (i) LiNGAM [Shimizu et al., 2006] (ii) PC [Spirtes and Glymour, 1991] and (iii) CCD [Richardson, 1996]; we call these three approaches fPCA-LiNGAM, fPCA-PC and fPCA-CCD respectively. Note that we did not use SEM-based cyclic discovery algorithm, LiNG-D [Lacerda et al., 2008] in the second step due to the unavailability of the code.

LiNGAM estimates a causal DAG based on the linear non-Gaussian assumption whereas PC generally returns only an equivalence class of DAGs based on conditional independence tests. CCD algorithm is a constraint-based causal discovery method, which yields an equivalence class of cyclic causal graphs. LiNGAM and PC are implemented in the pcalg package [Kalisch et al., 2018] in R. CCD algorithm is implemented in the py-tetrad [Ramsey et al., 2018] package in python.

Performance metrics. To assess the graph recovery performance, we calculated the true positive rate (TPR), false discovery rate (FDR), and Matthew's correlation coefficient (MCC). For TPR and MCC, higher is better, whereas lower FDR is better.

Results. Table 1 summarizes the results of 50 repeat simulations, demonstrating that the proposed FENCE model outperforms all competitors (fLiNG, fPCA-LiNGAM, and fPCA-CCD) across all combinations of n , p , and d . We provide the results of fPCA-PC in Section D of the Supplementary Materials, which are similar to those of fPCA-LiNGAM. We favored fPCA-PC and fPCA-CCD by counting a non-invariant edge between two nodes as a true positive as long as the two nodes are adjacent in the true graph. The superiority of FENCE is not unexpected for three reasons. First, fLiNG, fPCA-LiNGAM, and fPCA-PC are not specifically designed for learning cyclic graphs. Second, two-step approaches like fPCA-LiNGAM, fPCA-PC, and fPCA-CCD do not necessarily capture the causally embedded space through the functional principal components. Third, although fPCA-CCD can handle cyclic graphs, it, being a two-step approach, fails to capture true functional dependencies. Overall, these findings provide strong evidence of the effectiveness of FENCE compared to existing methods.

Additional simulations. We considered additional simulation scenarios with unevenly spaced grids, general exogenous variable distributions, true acyclic graphs and data generated using non-linear SEM, and also conducted sensitivity analyses of FENCE with respect to several hyperparameters; the results are presented in Section C of the Supplementary Materials. The performance of the proposed method is consistently better than competing methods and is relatively robust with respect to hyperparameter choices.

5 Real data application

Brain EEG data. We demonstrate the proposed FENCE model on a brain EEG dataset from an alcoholism study [Zhang et al., 1995]. This dataset was earlier used to demonstrate functional undirected graphical models [Zhu et al., 2016, Qiao et al., 2019] and functional Bayesian network [Zhou et al., 2022b]. Data were initially obtained from 64 electrodes placed on subjects' scalps, which captured EEG signals at 256 Hz (3.9 ms epoch) during a one-second period. The study consists of 122 subjects, out of which 77 are in the alcoholic group and 45 are in the control group. Each subject completed 120 trials. During each trial, the subject was exposed to either a single stimulus (a single picture) or two stimuli (a pair of pictures) shown on a computer monitor. We particularly focus on the EEG signals filtered at α frequency bands between 8 and 12.5Hz using the eegfilt function of the eeglab toolbox of Matlab as α band signals are associated with inhibitory control [Knyazev, 2007]. Given that the EEG measurements were recorded from each subject over multiple trials, these measurements are not independent of each other due to the time dependency of the trials. Moreover, since the measurements were obtained under various stimuli, the signals may have been affected by different stimulus effects. To mitigate these issues, we calculated the average of the band-filtered EEG signals for each subject across all trials under a single stimulus, resulting in a single event-related potential curve per electrode per subject. By doing so, we eliminated the potential dependence between the measurements and the influence of different stimulus types. We performed separate analyses of the two groups to identify both the similarities and dissimilarities in their brain effective connectivity.

We conducted a Shapiro-Wilk normality test on the observed functions for each of the p = 64 scalp positions at each of the m j = 256 ∀ j ∈ [ p ] time points to evaluate their Gaussianity. The results showed that for numerous combinations of scalp position and time point, the null hypothesis (which assumes that the observations are marginally Gaussian) was rejected. Thus, we conclude that the non-Gaussianity of the proposed model is appropriate. Next, for posterior inference, we ran MCMC for 20,000 iterations, discarded the first half as burn-in, and retained every 10th iteration after burn-in. The estimated causal graph by thresholding the posterior inclusion probability to 0 . 9 is given below in Figure 2.

Parietal

Frontal Lobe

Lobe

Fp1

Fpz

Fp2

pz

Fp2

F9

F7

AF7

AF3

AFZ4

AF4

AF8

8

F9

AF7

F3

AF4

AF8

F18

61

FT7

FC5

F3

FT8

(FT18)

Temoral

Lobe

Occipital

Lobe

FT9

FT7

F6

FT8

FT18

LPA

T7

C5

L6

18

RPA

RPA

TP9

TP7

CP5

CP3

CP1

CPz

CP2

CP4

CP6

TP8

(TP18)

6d

IP

TP8

TP18

P7

P5

P3

Pz

P2

P4

P6

P8

P9

P07

P03

POz

P04

P08

P10

P07

03

POz

P08

P10

01

0z

02

ZT

Results. There are some interesting findings. First, for both groups (alcoholic and control), brain regions that are located in adjacent positions tend to be more connected than the brain regions that are far apart. Second, dense connectivity is observed in the frontal region of the brain in both groups, with multiple cycles being formed. Third, compared to the control group, the alcoholic group has more connectivity across the left parietal and occipital lobes. Fourth, the same cycle of Iz, Cz, and RPA is observed in both groups.

Validity. We now discuss the validity of our real data results. In Hayden et al., 2007, it was observed that alcohol-dependent subjects exhibited frontal asymmetry, distinguishing them from the control group. Our own investigation aligns well with these results, as we have identified denser connectivity across various brain regions in the middle and left areas of the frontal lobe among alcoholic subjects, when compared to controls. Furthermore, Winterer et al., 2003 documented coherent differences between alcoholics and controls in the posterior hemispheres, specifically in the temporal, parietal, and occipital lobes. In accordance with their findings, our study provides additional support for this claim, as we have observed heightened activity with several cycles formed in those same regions within the alcoholic group when compared to the control group.

6 Discussion

We briefly highlight here several potential avenues for future development of our current work. First, an intriguing and important direction would be to explore the relaxation of the causal sufficiency assumption in the model identifiability. Second, our current model is based on a linear non-Gaussian assumption over the exogenous variables, but a nonlinear model could be considered as an alternative. Lastly, an alternative approach to determining the effective number of basis functions that span the lower-dimensional causal embedded space would be to utilize the ordered shrinkage priors [Bhattacharya and Dunson, 2011, Legramanti et al., 2020] in order to adaptively eliminate redundant components, resulting in a more flexible methodology.

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Peer reviews:
comments: SUMMARY:
The paper develops a causal discovery method for directed cyclic graph. The proposed method is a two-step approach which utilizes a Bayesian approach to reduce the dimension of functional data and performs the causal structure learning on the learned embeddings. Experiments on the synthetic and real-world datasets well demonstrate the effectiveness of the proposed method.

STRENGTHS:
1. There are seldom paper studying the causal discovery method on the directed cyclic graph. However, in some particular applications, there may exist cycles in the causal graphs.
2. The proposed method outperforms baselines in various settings.
3. The results of the brain EEG data provide some insights on the organization of brain regions.

WEAKNESSES:
1. It lacks some classical and SOTA baselines, e.g., NOTEARS[1], DAG-GNN[2], CSIvA[3]
[1] Zheng X, Aragam B, Ravikumar P K, et al. Dags with no tears: Continuous optimization for structure learning[J]. Advances in neural information processing systems, 2018, 31.
[2] Yu Y, Chen J, Gao T, et al. DAG-GNN: DAG structure learning with graph neural networks[C]//International Conference on Machine Learning. PMLR, 2019: 7154-7163.
[3] Learning to induce causal structure. ICLR 2023.
2. It is hard to set so many priors for the proposed model in practice.
3. The presentation of the manuscript could be improved. Fig.1 is hard to understand.
4. For the results of real data, it is better to also show the results of baselines.

QUESTIONS:
Please refer to weaknesses.

LIMITATIONS:
No
I have not found the limitations and  broader societal impacts of this manuscript.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-06-19

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper discusses the inference of causal relationships between multivariate functions through the proposal of a causal discovery model. The model involves embedding the functional nodes into a lower-dimensional space and separating the structural equation model into two components: the projection onto the space and its orthogonal complement in the Hilbert space defined on the domain. The identifiability of the proposed model is proven based on several assumptions, including disjoint cycles, as described in the Supplementary Materials. The paper presents a model inference method utilizing a fully Bayesian approach. Experimental results using simulated data demonstrate that the proposed method outperforms both conventional causal discovery methods for multivariable functional data and other conventional causal discovery methods. Additionally, insightful observations are made from the application of the proposed method to brain EEG data.\n\nSTRENGTHS:\nThe strongest aspect of this paper is its introduction of a causal discovery model for multivariate functional data, which accommodates the presence of cycles. This is a significant contribution since many multivariate functional datasets, such as EEG data, inherently involve cycles, including feedback loops. Additionally, the paper proves the identifiability of the model and presents a fully Bayesian approach to model inference. The effectiveness of the proposed method is demonstrated through the utilization of both simulated data and real-world EEG data.\n\nWEAKNESSES:\nThe discussion of the results of the brain EEG data is inadequate. While there are differences in connectivity between the alcoholic and control groups, readers are unable to determine the significance of these findings because most readers may not possess a background in brain science.\n\nThere may be a minor mistake: from my understanding, the subscript $j=1$ below the superscript $m_j$ in line 116 might actually be $u=1$.\n\nQUESTIONS:\nIs it possible to consider providing additional discussion on the results of the brain EEG data, in order to help readers ascertain the validity of the findings and comprehend their significance?\n\nLIMITATIONS:\nThe limitation of the proposed method lies in the assumptions required to establish the identifiability of the model.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a causal model, i.e. a linear structural equation model for multivariate functional data. The authors' proposed model does not require the dependency structure to conform to a DAG, allowing for modeling cyclic cause-effect relationships. Given certain assumptions on exogenous variables and cycle structure, the authors show the identifiability of their model; a central assumption of which is that the causal interrelations between the variables pertain only to a subset of the function space which the observations inhabit. The authors propose an MCMC sampling procedure to sample from the posterior of the parameters given prior hyperparameters. The authors test their model under various conditions in synthetic data, in addition to a real world application.\n\nSTRENGTHS:\n- The authors' research is soundly motivated, the central observation regarding the importance of being able to handle cyclic relationships in functional data is well-justified, especially given the application areas they consider.\n- The authors' central modeling assumption, that the causal interrelations pertain to a subset of the function space is interesting and has the potential to be utilized and expanded upon future research.\n- Minus some limiting assumptions, the authors' identifiability results and inference procedure is valuable.\n- The authors provide a rigorous exploration of various aspects of their model in comparison with baselines in synthetic data.\n\nWEAKNESSES:\n- The authors make a series of assumptions (1-6) in order to achieve identifiability, with the first one being causal sufficiency. Both the assumption of causal sufficiency and making simplifying assumptions in general is a widely used practice especially when expanding causal analysis to previously unexplored territory, and is usually acceptable to expedite the initial analysis of a new idea. Although it is still understandable that the authors make this assumption, for the kinds of data they are hoping to conduct causal analysis on this assumption unfortunately seems to be especially likely to be violated. For example, it sounds very improbable for measurements made in specific parts of the brain to not have latent confounding. A similar case against disjoint cycles can also be made, while the rest of the assumptions seem less offensive. That being said, I think the authors' contributions justify these limitations.\n- Given the presence of these limiting assumptions, some external validation through real data experiments is called for. The authors' experiment with real data present no external validation (e.g. discovering some known causal relationships among various brain regions).\n\nQUESTIONS:\n- In addition to the aforementioned assumptions, using a linear model is another disputable choice, as the authors note. Did the authors experiment with nonlinear generated data to examine the effects of this potential source of model misspecification?\n- Brain imaging is an obvious (and worthy) example for the utilization of such a modeling approach. Are there any other impactful application fields that can benefit from such work?\n- In L99, is it supposed to be $\\mathcal{H}_\\ell$ to $\\mathcal{H}_j$?\n- In L116, is it supposed to be $\\\\{(t_{ju}, X_{ju})\\\\}^{m_j}_{u=1}$?\n\nLIMITATIONS:\nThe authors generally transparent about their modeling assumptions and potential effects thereof. However some of the points I mentioned above would benefit from more elaboration in the final paper.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the author proposes an operator-based non-recursive linear structural equation based novel causal discovery framework that identifies causal relationships among functional objects in the presence of cycles and additional sampling noises. Furthermore, author demonstrates the effectiveness of proposed method from experiment and theory.\n\nSTRENGTHS:\nThe motivation is clear and the writing is well. The experiment results show that the proposed method significantly outperforms the baselines. The causal identifiability proof is impressive.\n\nWEAKNESSES:\nPlease refer to the Questions.\n\nQUESTIONS:\n1. The proposed year of your baselines fLiNG, LiNGAM, PC, CCD are 2022, 2006, 1991, 1996 respectively, more recent baselines should be considered. According to line 282 and 283, the codes of Lee (2022) and Yang (2022) are not available. However,  fLiNG’s code is also not public (I tried my best to find it but failed). Hence, how do you obtain the results reported in the paper ? \n2. In line 18, you said casual discovery is popular in machine learning. But both in introduction and related work, I do not find relative description to support this standpoint.\n3. How to obtain equation (3) ? It appears to be left-multiplying Equation (1) by  and  respectively. If so, the term in first row should be .\n4. In line 116, Why does j vary from 1 to  ? The range of j should be from 1 to p.\n5. Many symbols are used without definition, like Beta in line 241, Dir and IG in line 254. Some symbols are easy to search for their meanings, while others are not as straightforward.\n6. The equation (7) are not mentioned in Section 3, is it really necessary ?\n7. The name of Section 3 is Model inference. However, nothing but the prior distribution of parameters are introduced, this is confusing.\n8. Reference [Fangting Zhou, Kejun He, and Yang Ni. Causal discovery with heterogeneous observational data] mentions that they do not restrict their model to be acyclic, which is in conflict with your statement in line 56 (Related work section).\n\nLIMITATIONS:\nNo apparent negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the problem of learning causal structure from multivariate functional data that may involve cyclic interactions.\nIt employs an adaptive mapping to a lower dimensional space that retains the relevant causal information, in combination with a Bayesian framework that obtains posterior estimates through MCMC sampling. Effectiveness of the model is compared against several alternatives on synthetic and real-world data.\n\nSTRENGTHS:\nThe paper is exceptionally well written and contains a comprehensive and powerful approach to cyclic causal discovery.\n\nMany of the steps involved are not new by themselves, but brought together effectively to produce an elegant and novel method that shows promising performance. The descriptions and theoretical derivations are concise, clear and consistent, and each of the steps involved follows logically to reduce a challenging problem to an effective, fully Bayesian solution. I particularly liked the way the adaptive basis expansion in 3.2 was incorporated into the overall method.\n\nAssumptions are strong, but clearly and carefully spelled out, leading to the desired identifiability conclusion in Thm2.1.\n\nThe synthetic data experimental evaluation is convincing (although due to the required assumptions inevitably biased in favour of the proposed approach), and the alcoholic EEG application is interesting (though hard to evaluate for non-experts).\n\nWEAKNESSES:\n- although I really like the paper, I think the main contribution lies mostly in the way the various steps are brought together in a clear and coherent way than in fundamentally new insights or approaches. (Nevertheless, I do consider the end result a valuable and novel contribution.)\n- assumptions are quite strong: to (still) see causal sufficiency feature so prominently is disappointing, the ‘disjoint cycles’ should be unnecessary (as admitted by the authors), and ‘non-Gaussianity’ may seem generic but implies identifiability may rely on weak distributional signals (data hungry) and it excludes the challenging complications encountered in the linear Gaussian case that e.g. CCD was specifically designed to handle. \n\n- one confusing aspect (at first) was that in the beginning of the paper the introduction of the time measurements seemed to suggest we are doing time-series analysis, but on closer inspection it is actually closer to standard observational causal inference (with cycles), right?\n\n- some questionable claims, e.g. l.245 ‘strongly prevents false discoveries’ could equally be stated as ‘heavily biased towards sparsity’, and l.307 ‘strong evidence of the effectiveness of FENCE compared to existing methods’ should contain the caveat ‘under the stated assumptions 1-6’. In particular, starting from PCA for LINGAM / CCD seems questionable, where LiNGAM is designed for DAGs, and independence based methods like CCD do not (need to) rely on functional assumptions that are essential for FENCE.\n\nQUESTIONS:\n- what is the relation/difference between your model assumptions and e.g. the simple SCMs in Bongers et al. [‘Foundations of structural causal models with cycles and latent variables.’,2018]?\n\n- how do you determine Kj for the lower dimensional space embedding? \n\n- is it true that you essentially treat multiple observations at different time points as independent observations on different system instances in equilibrium?\n\n- in. Table 1, how do you compare non-invariant edges in the equivalence class of e.g. PC/CCD vs. the ground truth? (as ‘half wrong’?)\n- given the causal sufficiency assumption, how come you find bidirected edges in Fig.2?\n\nOther remarks (for lack of a better place to put them):\n- title seems grammatically a bit weird \n- 5 ‘enhance interpretability’ => I understand the practical effectiveness in going via a lower dimensional internal representation, but the output does not reflect this, right? So how does it affect ‘interpretability’?\n- 88 ‘cyclic component’ => this definition seems a bit off as it allows for vertices in a subgraph that are not part of a cycle (so only the full graph G can be a ‘maximal’ cyclic component). I would suggest using the standard term ‘strongly connected component’.\n- 258 ‘simulate posterior samples through MCMC’ => actually I would not mind seeing a bit more on this step in the main paper\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2023-07-12"}