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Functional Causal Discovery via Conditional Covariance Ordering

Keyu Li, Ruoxu Tan

arXiv:2609.27256v1stat.MEstat.ML

TL;DR

Causal direction discovery for functional variables is challenging without assumptions on functional form or error distribution. The paper uses conditional covariance norms to order variables and mixed regression to recover edges, establishing asymptotic consistency and reporting satisfactory simulations and literature-consistent EEG connectivity.

  • Problem

    Identifying causal directions among functional variables without assuming a specific functional form or error distribution remains a central challenge.

    Source: §1 · p. 1

  • Method

    The procedure orders variables using conditional covariance norms, then recovers edges with mixed linear–nonlinear regression and variable selection.

    Source: §1 · p. 3, §7 · p. 27

  • Results

    The authors establish asymptotic consistency for ordering, regression, and DAG identification; simulations perform satisfactorily against existing methods, and EEG connectivity agrees with related literature.

    Source: §1 · p. 3

  • Takeaways & Limitations

    In an EEG application, inferred brain connectivity is consistent with related literature.

    Source: §1 · p. 3

    The asymptotic theory treats the number of nodes p as fixed.

    Source: §4 · p. 15

Abstract

from arXiv · show

We study causal discovery where each node is a random function. Previous studies on this topic rely on structural assumptions, e.g., linearity or non-linearity, and distributional assumptions, e.g., Gaussianity or non-Gaussianity. In contrast, we make use of covariance operators to avoid these assumptions. Under functional additive noise models, we propose a new sufficient condition to identify a valid topological ordering based on comparing norms of conditional covariance operators. Taking advantage of this identifiability condition, we develop a new mixed regression model that subsumes linear and non-linear models. Together with variable selection, our procedure yields an estimation of the causal directed acyclic graph (DAG) for functional variables. In theory, we develop the least-squares-type theory of this regression model, and derive asymptotic consistency of order determination, sparse regression, as well as identifying the DAG. Computational algorithms based on discrete observations are provided. Applied to simulated data, our approach performs satisfactorily among existing approaches. A real data example of brain effective connectivity is also presented.

1 Introduction

The paper addresses causal direction discovery for functional variables without requiring a specific functional form or error distribution. It proposes conditional-covariance ordering and mixed regression to estimate functional causal DAGs.

1 Introduction

  • The method targets causal directions among functional variables without prespecifying their functional form or error distribution.Existing functional causal-discovery approaches use varied modeling assumptions, including nonlinear Gaussian or linear non-Gaussian models (Yang et al., 2024).Source: §1 · pp. 1–2
  • Comparing norms of integrated conditional covariance operators yields a sufficient condition for recovering a valid topological order, accommodating linear or nonlinear models and Gaussian or non-Gaussian errors.The condition is strictly more general than bounded linear models with equal error-covariance norms and non-degenerate edges.Source: §1 · pp. 2–3
  • A mixed regression framework combines linear and nonlinear components, then uses variable selection to identify edges after estimating the ordering.The paper adapts a group SCAD penalty for variable selection.Source: §1 · p. 3
  • The authors establish asymptotic consistency for ordering, regression, and DAG identification; simulations perform satisfactorily against existing methods, and EEG connectivity agrees with related literature.The article also presents a brain effective-connectivity application.Source: §1 · p. 3

2 Functional Additive Noise Model and Its Identifia-

The paper defines functional additive noise models and uses conditional covariance operators to characterize a sufficient ordering condition for causal identification. Sequentially minimizing conditional covariance norms estimates the order, after which sparse regression can select edges.

2.1 Model and Notations

  • FANM specifies nodewise operator effects and mutually independent, mean-zero random-function errors; its integrated conditional covariance operator summarizes residual variation given selected functions.The conditional covariance operator is defined for each function conditional on a subset of the other functions.Source: §2.1 · pp. 4–5

2.2 Identifiability via Conditional Covariance Operators

  • The order is obtained by sequentially choosing the remaining node with the smallest conditional HS norm, with deterministic tie-breaking when needed.After order determination, sparse regression is used for edge selection.Source: §2.2 · pp. 5, 6, 8
  • Under bounded linear effects, equal error-covariance HS norms, and non-degenerate edges, a parent-complete node has a smaller conditional HS norm than each of its descendants.The proposition applies to any ancestral conditioning set containing the node’s parents.Source: §2.2 · p. 7
  • The paper’s ordering condition requires no specific structural form, error distribution, or error-covariance norm, and can identify models with unequal error-covariance norms.The equal-norm bounded-linear setting with non-degenerate edges implies the condition, but the converse does not hold.Source: §2.2 · pp. 7–8

3 Functional Mixed Additive Noise Model

FMANM combines linear and nonlinear components for functional causal models; the section develops its regression foundations and an estimation procedure.

3.1 Model and Theoretical Foundation

  • FMANM includes both linear and nonlinear components, with its nonlinear component built on a nested RKHS representation of functional features.The model uses mutually independent, mean-zero functional errors.Source: §3.1 · pp. 8–9
  • Regressing a function on its parents and non-descendants yields nonzero regression components only for its parents, motivating function-on-function regression for edge identification.The model’s independent errors make it Markovian: each variable is conditionally independent of its non-descendants given its parents.Source: §3.1 · p. 11
  • The least-squares-type objective unifies linear and nonlinear dependencies, and its minimizer is the regression operator under the stated assumptions.Its quadratic formulation requires inversion of a unified covariance operator that captures coupling between original variables and kernel embeddings.Source: §3.1 · p. 12
  • Variable selection identifies edges by selecting nonzero coefficient blocks, where either the linear component, nonlinear component, or both may be nonzero.The estimation procedure uses a SCAD penalty to encourage sparse regression coefficients.Source: §3.1 · p. 12, §3.2 · p. 14

3.2 Estimation

  • For estimation, coefficients are restricted to finite-dimensional spaces spanned by the first d empirical eigenfunctions, and a local minimizer can satisfy estimation and selection consistency.The SCAD-penalized criterion is nonconvex, so the minimizer need not be unique.Source: §3.2 · pp. 14–15

4 Asymptotic Theory

The asymptotic theory establishes consistency for ordering and regression, yielding consistency of the two-stage DAG-identification procedure under stated assumptions. Its results treat the number of nodes as fixed and, without Gaussianity, use bounded fourth moments, which leads to polynomial tail probabilities.

4 Asymptotic Theory

  • The theory establishes uniform consistency of order determination and regression-coefficient convergence rates, retaining graph size p in the convergence statements.The number of nodes is fixed because the proof of the ordering theorem uses mathematical induction; diverging p would require additional conditions.Source: §4 · p. 15
  • The estimated ordering is a valid topological ordering with probability tending to one when the ordering error bound E_n,1 tends to zero.The greedy procedure may encounter ties, so the result does not single out a particular ordering.Source: §4 · p. 16
  • The regression theory requires covariance eigenvalue separation and smoothness conditions, with truncation bias negligible relative to the specified sequence c_d.The assumptions also require the truncated unified covariance operator to be positive definite on the truncation subspace.Source: §4 · pp. 16–17
  • Consistency of SCAD-penalized regression includes a local minimizer with estimation and variable-selection consistency, under the theorem’s assumptions and rate conditions.The model includes a non-linear component, which the authors say makes the technical arguments more involved than in the function-on-function linear model.Source: §4 · p. 17
  • Under Assumptions 1–5 and the stated rate conditions, the two-stage procedure consistently identifies the DAG as sample size grows.The result combines consistency of order determination with estimation and selection consistency for SCAD-penalized regression.Source: §4 · pp. 17–18
  • Without Gaussianity, the theory assumes bounded fourth moments, yielding polynomial tail probabilities rather than the exponential tails available under Gaussian or sub-Gaussian assumptions.The authors state that exponential tails relax the rate conditions for theorems on ordering and regression.Source: §4 · p. 18

5 Practical Computation from Discrete Data

The discrete-data procedure represents observed curves with finite-dimensional coordinates, constructs linear and nonlinear components, then uses an ordering-based recursive selection and sparse regression to estimate edges. Its computational choices are not incorporated into the continuous-L2 theory.

5 Practical Computation from Discrete Data

  • Observed curves are smoothed onto a common grid when sampling times differ, then represented by finite-dimensional coordinates and principal-component scores.The number of components is selected to explain at least 92% of empirical variance at each node.Source: §5 · p. 18
  • The nonlinear component uses a kernel Gram matrix built from the principal-component scores and a centered eigendecomposition.The implementation uses RKHS reconstructions and data-adaptive truncation as computational surrogates.Source: §5 · pp. 18–19
  • The procedure recursively selects an ordering and uses SCAD-penalized block regression for edge selection, with nonzero fitted blocks defining edges.It uses SCAD with a = 6, selects λ by a BIC-type rule, and computes a numerical solution by block proximal-gradient iterations.Source: §5 · p. 19
  • The default ϵ = n^-2/5 is a computational heuristic not covered by Theorem 2’s sufficient rate conditions.The continuous-L2 theory also does not incorporate reconstruction, truncation, smoothing, or discretization errors.Source: §5 · p. 19

6 Numerical Experiments

Simulations favor the proposed methods across nonlinear settings, with FMixedANM often among the strongest; the EEG analysis finds group-specific connectivity patterns, including alcohol-specific edges concentrated in parietal electrodes.

6.1 Simulation

  • Simulations compare FMixedANM with purely linear FLinANM and purely nonlinear FNonANM variants across graph sizes p=20, 40, and 60 and sample sizes n=200 and 400; each scenario has 100 Monte Carlo replicates.The j<i restriction is used only to generate DAGs; estimation procedures are not supplied with the topological ordering.Source: §6.1 · p. 20
  • For linear models, FMixedANM leads F1, closely followed by FLinANM, while FLNGB has the best SHD and the three proposed methods follow.FNG performs worst on F1 under linear models.Source: Figure 1 · p. 21, §6.1 · p. 24
  • Across nonlinear models (iii)–(v), all proposed methods outperform FNG on both SHD and F1, while FMixedANM almost always ranks among the best two and is best in some settings.Model (v) violates condition (2.2), so these results also indicate robustness when the proposed identifiability condition fails.Source: Figure 2 · p. 22, Figure 3 · p. 23, §6.1 · p. 24

6.2 Brain effective connectivity analysis

  • Across methods, parietal electrodes are a major source of alcohol-specific edges for the three proposed methods and FLNGB, whereas FNG concentrates these edges at frontal-pole electrodes.For FMixedANM, FLinANM, and FNonANM, the parietal counts are 12 of 26, 15 of 58, and 10 of 24, respectively.Source: §6.2 · p. 25
  • FNG and FLNGB place some estimated edges in occipital regions and non-cortical channels.The non-cortical channels are X, Y, and nd.Source: §6.2 · p. 26
  • In the FMixedANM EEG comparison, 26 edges involving 35 electrodes are alcoholic-group-specific, while 14 edges involving 21 electrodes are control-group-specific.The alcoholic and control groups share 10 edges among their respective 36 and 24 estimated edges.Source: §6.2 · pp. 26–27, Figure 4 · p. 26
  • Of the 26 alcohol-specific edges, 12 originate from parietal electrodes; 7 of 14 control-specific edges originate from frontal electrodes, with the rest in parietal and occipital regions.The regional patterns are described as consistent with prior EEG studies, but this single small-sample analysis does not validate inferred causal directions.Source: §6.2 · p. 27

7 Discussion

The discussion recaps a two-stage functional causal discovery procedure and its consistency results, then identifies limits in the theory and directions for extending and validating the framework.

7 Discussion

  • The procedure estimates an ordering by sequentially minimizing conditional covariance norms, then recovers edges with SCAD-penalized function-on-function regression.The identifiability condition avoids specific structural or distributional assumptions, and the regression includes linear and nonlinear components.Source: §7 · p. 27
  • The theory establishes consistency for order determination, regression-coefficient estimation, and edge recovery, but does not account for smoothing or discretization errors.The asymptotic analysis treats the number of nodes p as fixed.Source: §7 · p. 27
  • For regression alone, FMANM conceptually permits diverging p, but rigorous theoretical justification for that setting remains open.The current theory treats p as fixed because the proof of Theorem 1 uses mathematical induction for finite p.Source: §7 · p. 28
  • Future extensions include joint estimation of related functional DAGs, mixed functional and vector-valued data, and incorporating scientific knowledge or interventional data.Independent-dataset or experimental validation is also proposed to strengthen practical relevance and interpretation.Source: §7 · p. 28
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