Functional Graphical Models

ABSTRACT Graphical models have attracted increasing attention in recent years, especially in settings involving high-dimensional data. In particular, Gaussian graphical models are used to model the conditional dependence structure among multiple Gaussian random variables. As a result of its computational efficiency, the graphical lasso (glasso) has become one of the most popular approaches for fitting high-dimensional graphical models. In this paper, we extend the graphical models concept to model the conditional dependence structure among p random functions. In this setting, not only is p large, but each function is itself a high-dimensional object, posing an additional level of statistical and computational complexity. We develop an extension of the glasso criterion (fglasso), which estimates the functional graphical model by imposing a block sparsity constraint on the precision matrix, via a group lasso penalty. The fglasso criterion can be optimized using an efficient block coordinate descent algorithm. We establish the concentration inequalities of the estimates, which guarantee the desirable graph support recovery property, that is, with probability tending to one, the fglasso will correctly identify the true conditional dependence structure. Finally, we show that the fglasso significantly outperforms possible competing methods through both simulations and an analysis of a real-world electroencephalography dataset comparing alcoholic and nonalcoholic patients.

• Publication year

  2018

• Venue

  Journal of the American Statistical Association

• Publication date

  2018-08-15

• Fields of study

  Mathematics, Computer Science

• Identifiers
  DOI 10.1080/01621459.2017.1390466
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Functional Data Analysis

  2016cited by this paper
• Theoretical foundations of functional data analysis, with an introduction to linear operators

  2015cited by this paper
• Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains

  2015cited by this paper
• Domain-Adversarial Training of Neural Networks

  2015cited by this paper
• Sparse precision matrix estimation via lasso penalized D-trace loss

  2014cited by this paper
• Bayesian Graphical Models for Multivariate Functional Data

  2014cited by this paper
• Multivariate functional principal component analysis: A normalization approach

  2014cited by this paper
• Structural Pursuit Over Multiple Undirected Graphs

  2014influential reference
• Concentration Inequalities: A Nonasymptotic Theory of Independence

  2013cited by this paper
• Joint estimation of multiple graphical models from high dimensional time series

  2013cited by this paper
• Regularized matrix regression

  2012cited by this paper
• Sparse matrix inversion with scaled Lasso

  2012cited by this paper
• The joint graphical lasso for inverse covariance estimation across multiple classes

  2011cited by this paper
• Matrix variate logistic regression model with application to EEG data.

  2011cited by this paper
• Exact Covariance Thresholding into Connected Components for Large-Scale Graphical Lasso

  2011cited by this paper
• The Graphical Lasso: New Insights and Alternatives

  2011cited by this paper
• On Time Varying Undirected Graphs

  2011cited by this paper
• Joint estimation of multiple graphical models.

  2011cited by this paper
• A Constrained ℓ1 Minimization Approach to Sparse Precision Matrix Estimation

  2011cited by this paper
• Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers

  2011influential reference
• New Insights and Faster Computations for the Graphical Lasso

  2011cited by this paper
• A note on the group lasso and a sparse group lasso

  2010cited by this paper
• On dimension folding of matrix- or array-valued statistical objects

  2010cited by this paper
• High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence

  2008influential reference
• Stability selection

  2008cited by this paper
• Time varying undirected graphs

  2008cited by this paper
• Sparse inverse covariance estimation with the graphical lasso.

  2008cited by this paper
• Partial Correlation Estimation by Joint Sparse Regression Models

  2008cited by this paper
• Motivation, emotion, and their inhibitory control mirrored in brain oscillations.

  2007cited by this paper
• Model selection and estimation in the Gaussian graphical model

  2007influential reference
• Sparsistency and Rates of Convergence in Large Covariance Matrix Estimation.

  2007cited by this paper
• Model selection and estimation in regression with grouped variables

  2006cited by this paper
• Patterns of regional brain activity in alcohol-dependent subjects.

  2006influential reference
• High-dimensional graphs and variable selection with the Lasso

  2006cited by this paper
• Sparsity and smoothness via the fused lasso

  2005cited by this paper
• Functional Data Analysis for Sparse Longitudinal Data

  2005cited by this paper
• Significance analysis of time course microarray experiments.

  2005cited by this paper
• The Dantzig selector: Statistical estimation when P is much larger than n

  2005influential reference
• Functional Graphical Models

  2003cited by this paper
• Generic Estimation Procedures for Orientation with Minimum and Redundant Information

  2001cited by this paper
• Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization

  2001influential reference
• Model Selection Based on Minimum Description Length.

  2000cited by this paper
• Distance between Attributed Graphs and Function-Described Graphs Relaxing 2nd Order Restrictions

  2000cited by this paper
• Linear Processes in Function Spaces

  2000cited by this paper
• Linear Processes in Function Spaces: Theory And Applications

  2000cited by this paper
• Principal component models for sparse functional data

  1999cited by this paper
• On Computing the Subset Graph of a Collection of Sets

  1999cited by this paper
• Statistical mechanics of neocortical interactions: Canonical momenta indicatorsof electroencephalography

  1997cited by this paper
• A Hypergraph Framework for Optimal Model-Based Decomposition of Design Problems

  1997cited by this paper
• Regression Shrinkage and Selection via the Lasso

  1996cited by this paper
• Parametric Feature Detection

  1996cited by this paper
• Statistical mechanics of neocortical interactions : Canonical momenta indicators of electroencephalography

  1995cited by this paper
• Event related potentials during object recognition tasks.

  1995cited by this paper
• Three-dimensional computer vision: a geometric viewpoint

  1993cited by this paper
• Estimating the mean and covariance structure nonparametrically when the data are curves

  1991cited by this paper
• Reliability analysis of parameter estimation in linear models with applications to mensuration problems in computer vision

  1987cited by this paper
• A UNIVERSAL PRIOR FOR INTEGERS AND ESTIMATION BY MINIMUM DESCRIPTION LENGTH

  1983cited by this paper

• A neighbour selection approach for identifying differential networks in conditional functional graphical models

  2026cites this paper
• Monitoring Covariance in Multichannel Profiles via Functional Graphical Models

  2026cites this paper
• Learning Functional Graphs with Nonlinear Sufficient Dimension Reduction

  2026influential citation
• Compositional data graphical model learning based on conditional independence measurement

  2026cites this paper
• Trans-Glasso: A Transfer Learning Approach to Precision Matrix Estimation

  2026cites this paper
• Graphical model for mixed data types

  2025cites this paper
• Simultaneous clustering and optimization in function-on-scalar regression

  2025cites this paper
• A framework for analyzing EEG data using high-dimensional tests

  2025cites this paper
• Predicting graphical models based on functional data

  2025cites this paper
• Covariate-dependent Graphical Model Estimation via Neural Networks with Statistical Guarantees

  2025cites this paper
• Understanding multimorbidity: insights with graphical models

  2025cites this paper
• Variable selection of multiple types of data: a PLS approach

  2025cites this paper
• Causal discovery and fault diagnosis based on mixed data types for system reliability modeling

  2025cites this paper
• Finite mixtures of functional graphical models: Uncovering heterogeneous dependencies in high-dimensional data

  2025cites this paper
• Systemic Competitiveness in the EU Cereal Value Chain: A Network Perspective for Policy Alignment

  2025cites this paper
• Gaussian Graphical Models for Partially Observed Multivariate Functional Data

  2025influential citation
• Distribution-valued data graphical model estimation based on M-LDQ feature embedding

  2025cites this paper
• Equivalence Test for Mean Functions from Multi-population Functional Data

  2025cites this paper
• Deep Neural Network for Functional Graphical Models Structure Learning

  2025cites this paper
• Bayesian functional graphical models with change-point detection

  2024cites this paper
• A Copula Graphical Model for Multi-Attribute Data using Optimal Transport

  2024cites this paper
• Functional Gaussian Graphical Regression Models For Air Quality Data

  2024influential citation
• Functional-Edged Network Modeling

  2024cites this paper
• Functional linear non-Gaussian acyclic model for causal discovery

  2024cites this paper
• Graphical Principal Component Analysis of Multivariate Functional Time Series

  2024cites this paper
• Trans-Glasso: A Transfer Learning Approach to Precision Matrix Estimation

  2024cites this paper
• Functional Directed Acyclic Graphs

  2024influential citation
• IDGM: an approach to estimate the graphical model of interval-valued data

  2024influential citation
• A comprehensive survey of recent research on profile data analysis

  2024cites this paper
• Bayesian nonparametric mixtures of categorical directed graphs for heterogeneous causal inference

  2024cites this paper
• Mean and covariance estimation for discretely observed high-dimensional functional data: Rates of convergence and division of observational regimes

  2024cites this paper
• Comparing fundraising campaigns in healthcare using psychophysiological data: a network-based approach

  2024cites this paper
• Variational Nonparametric Inference in Functional Stochastic Block Model

  2024cites this paper
• Identifying stationary microbial interaction networks based on irregularly spaced longitudinal 16S rRNA gene sequencing data

  2024cites this paper
• Optimal Linear Discriminant Analysis for High-Dimensional Functional Data

  2023cites this paper
• Multi-view metro station clustering based on passenger flows: a functional data-edged network community detection approach

  2023cites this paper
• Second-Generation Functional Data

  2023cites this paper
• High-dimensional rank-based graphical models for non-Gaussian functional data

  2023influential citation
• From sparse to dense functional data in high dimensions: Revisiting phase transitions from a non-asymptotic perspective

  2023cites this paper
• The functional graphical lasso

  2023influential citation
• On Sufficient Graphical Models

  2023cites this paper
• Group Integrative Dynamic Factor Models With Application to Multiple Subject Brain Connectivity

  2023cites this paper
• Structural learning of functional directed graphical models with incomplete signals

  2023cites this paper
• Latent factor model for multivariate functional data

  2023influential citation
• Variable Selection and Minimax Prediction in High-dimensional Functional Linear Models

  2023cites this paper
• Nonlinear directed acyclic graph estimation based on the kernel partial correlation coefficient

  2023cites this paper
• Directed Cyclic Graph for Causal Discovery from Multivariate Functional Data

  2023cites this paper
• Factor-guided estimation of large covariance matrix function with conditional functional sparsity

  2023cites this paper
• Group integrative dynamic factor models for inter- and intra-subject brain networks

  2023cites this paper
• Continuously indexed graphical models

  2023cites this paper
• Functional structural equation model

  2022cites this paper
• Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions

  2022influential citation
• Linear Manifold Modeling and Graph Estimation based on Multivariate Functional Data with Different Coarseness Scales

  2022influential citation
• A joint latent factor analyzer and functional subspace model for clustering multivariate functional data

  2022cites this paper
• Graph-constrained analysis for multivariate functional data

  2022cites this paper
• The Functional LiNGAM

  2022cites this paper
• Functional Bayesian networks for discovering causality from multivariate functional data

  2022influential citation
• Functional mixtures-of-experts

  2022cites this paper
• On principal graphical models with application to gene network

  2022cites this paper
• Latent Multimodal Functional Graphical Model Estimation

  2022influential citation
• Mixture Of Functional Graphical Models

  2022influential citation
• Bayesian Inferences on Neural Activity in EEG-Based Brain-Computer Interface

  2022cites this paper
• Bayesian Regularization for Functional Graphical Models with Applications to Neuroimaging

  2021influential citation
• Network Modeling in Biology: Statistical Methods for Gene and Brain Networks.

  2021cites this paper
• Conditional independence testing in Hilbert spaces with applications to functional data analysis

  2021cites this paper
• Nonparametric and high-dimensional functional graphical models

  2021influential citation
• High-dimensional Functional Graphical Model Structure Learning via Neighborhood Selection Approach

  2021influential citation
• Modeling Massive Highly Multivariate Nonstationary Spatial Data with the Basis Graphical Lasso

  2021cites this paper
• Conditional Functional Graphical Models

  2021influential citation
• Gaussian graphical modeling for spectrometric data analysis

  2021cites this paper
• Monitoring Heterogeneous Multivariate Profiles Based on Heterogeneous Graphical Model

  2021influential citation
• Challenges of modeling and analysis in cybermanufacturing: a review from a machine learning and computation perspective

  2021cites this paper
• Bayesian functional graphical models

  2021cites this paper
• Statistical inference for the slope parameter in functional linear regression

  2021cites this paper
• Joint Functional Gaussian Graphical Models

  2021cites this paper
• From multivariate to functional data analysis: Fundamentals, recent developments, and emerging areas

  2021cites this paper
• Temporal Point Process Graphical Models

  2021cites this paper
• Nonparametric Functional Graphical Modeling Through Functional Additive Regression Operator

  2021cites this paper
• FuDGE: Functional Differential Graph Estimation with fully and discretely observed curves

  2020influential citation
• The Sylvester Graphical Lasso (SyGlasso)

  2020cites this paper
• Doubly functional graphical models in high dimensions

  2020cites this paper
• FuDGE: A Method to Estimate a Functional Differential Graph in a High-Dimensional Setting

  2020influential citation
• Finite sample theory for high-dimensional functional/scalar time series with applications

  2020cites this paper
• A Nonconvex Framework for Structured Dynamic Covariance Recovery

  2020cites this paper
• Sparse Functional Principal Component Analysis in High Dimensions.

  2020cites this paper
• Copula Gaussian Graphical Models for Functional Data

  2020influential citation
• Functional directed graphical models and applications in root-cause analysis and diagnosis

  2020cites this paper
• Online Structural Change-Point Detection of High-dimensional Streaming Data via Dynamic Sparse Subspace Learning

  2020cites this paper
• A Wavelet-Based Independence Test for Functional Data With an Application to MEG Functional Connectivity

  2020cites this paper
• Brain Network Constancy and Participant Recognition: an Integrated Approach to Big Data and Complex Network Analysis

  2020cites this paper
• Real-time fault detection and process control based on multi-channel sensor data fusion

  2020cites this paper
• Modeling and Analyzing Neural Signals with Phase Variability using the Fisher-Rao Registration

  2020cites this paper
• A New Perspective on Dependence in High-Dimensional Functional/Scalar Time Series: Finite Sample Theory and Applications

  2020influential citation
• On consistency and sparsity for high-dimensional functional time series with application to autoregressions

  2020cites this paper
• Statistical inference for high-dimensional panel functional time series

  2020cites this paper
• Weighted empirical likelihood inference for dynamical correlations

  2019cites this paper
• Direct Estimation of Differential Functional Graphical Models

  2019influential citation
• Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes

  2019influential citation
• Dynamic Multivariate Functional Data Modeling via Sparse Subspace Learning

  2018cites this paper
• Fast covariance estimation for multivariate sparse functional data

  2018cites this paper