### Abstract

We introduce a nonparametric graphical model whose observations on vertices are functions. Many modern applications, such as electroencephalogram and functional magnetic resonance imaging (fMRI), produce data are of this type. The model is based on additive conditional independence (ACI), a statistical relation that captures the spirit of conditional independence without resorting to multi-dimensional kernels. The random functions are assumed to reside in a Hilbert space. No distributional assumption is imposed on the random functions: instead, their statistical relations are characterized nonparametrically by a second Hilbert space, which is a reproducing kernel Hilbert space whose kernel is determined by the inner product of the first Hilbert space. A precision operator is then constructed based on the second space, which characterizes ACI, and hence also the graph. The resulting estimator is relatively easy to compute, requiring no iterative optimization or inversion of large matrices. We establish the consistency and the convergence rate of the estimator. Through simulation studies we demonstrate that the estimator performs better than the functional Gaussian graphical model when the relations among vertices are nonlinear or heteroscedastic. The method is applied to an fMRI dataset to construct brain networks for patients with attention-deficit/hyperactivity disorder. Supplementary materials for this article are available online.

Original language | English (US) |
---|---|

Pages (from-to) | 1637-1655 |

Number of pages | 19 |

Journal | Journal of the American Statistical Association |

Volume | 113 |

Issue number | 524 |

DOIs | |

State | Published - Oct 2 2018 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

}

*Journal of the American Statistical Association*, vol. 113, no. 524, pp. 1637-1655. https://doi.org/10.1080/01621459.2017.1356726

**A Nonparametric Graphical Model for Functional Data With Application to Brain Networks Based on fMRI.** / Li, Bing; Solea, Eftychia.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A Nonparametric Graphical Model for Functional Data With Application to Brain Networks Based on fMRI

AU - Li, Bing

AU - Solea, Eftychia

PY - 2018/10/2

Y1 - 2018/10/2

N2 - We introduce a nonparametric graphical model whose observations on vertices are functions. Many modern applications, such as electroencephalogram and functional magnetic resonance imaging (fMRI), produce data are of this type. The model is based on additive conditional independence (ACI), a statistical relation that captures the spirit of conditional independence without resorting to multi-dimensional kernels. The random functions are assumed to reside in a Hilbert space. No distributional assumption is imposed on the random functions: instead, their statistical relations are characterized nonparametrically by a second Hilbert space, which is a reproducing kernel Hilbert space whose kernel is determined by the inner product of the first Hilbert space. A precision operator is then constructed based on the second space, which characterizes ACI, and hence also the graph. The resulting estimator is relatively easy to compute, requiring no iterative optimization or inversion of large matrices. We establish the consistency and the convergence rate of the estimator. Through simulation studies we demonstrate that the estimator performs better than the functional Gaussian graphical model when the relations among vertices are nonlinear or heteroscedastic. The method is applied to an fMRI dataset to construct brain networks for patients with attention-deficit/hyperactivity disorder. Supplementary materials for this article are available online.

AB - We introduce a nonparametric graphical model whose observations on vertices are functions. Many modern applications, such as electroencephalogram and functional magnetic resonance imaging (fMRI), produce data are of this type. The model is based on additive conditional independence (ACI), a statistical relation that captures the spirit of conditional independence without resorting to multi-dimensional kernels. The random functions are assumed to reside in a Hilbert space. No distributional assumption is imposed on the random functions: instead, their statistical relations are characterized nonparametrically by a second Hilbert space, which is a reproducing kernel Hilbert space whose kernel is determined by the inner product of the first Hilbert space. A precision operator is then constructed based on the second space, which characterizes ACI, and hence also the graph. The resulting estimator is relatively easy to compute, requiring no iterative optimization or inversion of large matrices. We establish the consistency and the convergence rate of the estimator. Through simulation studies we demonstrate that the estimator performs better than the functional Gaussian graphical model when the relations among vertices are nonlinear or heteroscedastic. The method is applied to an fMRI dataset to construct brain networks for patients with attention-deficit/hyperactivity disorder. Supplementary materials for this article are available online.

UR - http://www.scopus.com/inward/record.url?scp=85057300753&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057300753&partnerID=8YFLogxK

U2 - 10.1080/01621459.2017.1356726

DO - 10.1080/01621459.2017.1356726

M3 - Article

AN - SCOPUS:85057300753

VL - 113

SP - 1637

EP - 1655

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 524

ER -