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Hierarchical Spatio-Temporal Modeling of Resting State fMRI Data

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Studies in Neural Data Science (START UP RESEARCH 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 257))

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Abstract

In recent years, state of the art brain imaging techniques like Functional Magnetic Resonance Imaging (fMRI), have raised new challenges to the statistical community, which is asked to provide new frameworks for modeling and data analysis. Here, motivated by resting state fMRI data, which can be seen as a collection of spatially dependent functional observations among brain regions, we propose a parsimonious but flexible representation of their dependence structure leveraging a Bayesian time-dependent latent factor model. Adopting an assumption of separability of the covariance structure in space and time, we are able to substantially reduce the computational cost and, at the same time, provide interpretable results. Theoretical properties of the model along with identifiability conditions are discussed. For model fitting, we propose a mcmc algorithm to enable posterior inference. We illustrate our work through an application to a dataset coming from the enkirs project, discussing the estimated covariance structure and also performing model selection along with network analysis. Our modeling is preliminary but offers ideas for developing fully Bayesian fMRI models, incorporating a plausible space and time dependence structure.

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Notes

  1. 1.

    Richer modeling might allow heterogeneity in variances, e.g., across regions but we do not consider that here.

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Acknowledgements

We are grateful to Greg Kiar and Eric Bridgeford from NeuroData at Johns Hopkins University, who graciously pre-processed the raw DTI and R-fMRI imaging data available at http://fcon_1000.projects.nitrc.org/indi/CoRR/html/nki_1.html, using the pipelines ndmg and c-pac. The authors are also thankful to the organizers of StartUp Research for coordinating such a stimulating event.

Author information

Authors and Affiliations

  1. Department of Statistical Sciences, Sapienza University of Rome, Rome, Italy

    Alessia Caponera

  2. Department of Statistics and Quantitative Methods, University of Milano-Bicocca, Milan, Italy

    Francesco Denti

  3. Department of Decision Sciences, Bocconi University, Milan, Italy

    Tommaso Rigon

  4. Department of Statistical Sciences, University of Padova, Padua, Italy

    Andrea Sottosanti

  5. Department of Statistical Science, Duke University, Durham, North Carolina, USA

    Alan Gelfand

Authors
  1. Alessia Caponera
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  2. Francesco Denti
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  5. Alan Gelfand
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Corresponding author

Correspondence to Tommaso Rigon .

Editor information

Editors and Affiliations

  1. Department of Statistical Sciences, University of Padova, Padua, Italy

    Antonio Canale

  2. Department of Decision Sciences, Bocconi University, Milan, Italy

    Daniele Durante

  3. Department of Statistical Sciences, Università Cattolica del Sacro Cuore, Milan, Italy

    Lucia Paci

  4. Department of Statistical Sciences, University of Padova, Padua, Italy

    Bruno Scarpa

7 Computational Details

7 Computational Details

In this appendix we describe a simple Metropolis-Hastings for posterior inference. The algorithm is summarized in Algorithm 1. Additionally, we derive the identity in Eq. (15). Because of orthogonality, we have that \(\varvec{U}_{\varvec{T}}\varvec{U}_{\varvec{T}}^\mathsf {T} = I_{T \times T}\) and \(\varvec{U}_{\varvec{A}}\varvec{U}_{\varvec{A}}^\mathsf {T} = I_{L \times L}\) and recall also that the matrices \(\varvec{\varLambda }_{\varvec{T}}\) and \(\varvec{\varLambda }_{\varvec{A}}\) are diagonal, containing the eigenvalues of \(\varvec{\varSigma }_{\varvec{T}}\) and \(\varvec{\varSigma }_{\varvec{A}}\), respectively. Exploiting the spectral decompositions of \(\varvec{\varSigma }_{\varvec{T}}\) and \(\varvec{\varSigma }_{\varvec{A}}\) and the basic properties of the Kronecker product, we get

$$\begin{aligned} \begin{aligned} \varvec{C}&= (\varvec{U}_{\varvec{T}} \varvec{\varLambda }_{\varvec{T}} \varvec{U}_{\varvec{T}}^\mathsf {T})\otimes (\varvec{U}_{\varvec{A}} \varvec{\varLambda }_{\varvec{A}} \varvec{U}_{\varvec{A}}^\mathsf {T}) +\sigma ^2I_{n\times n} \\&= (\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})(\varvec{\varLambda }_{\varvec{T}} \otimes \varvec{\varLambda }_{\varvec{A}})(\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})^\mathsf {T} +\sigma ^2I_{n\times n}. \end{aligned} \end{aligned}$$

Then, we can write the identity matrix \(I_{n \times n} = (\varvec{U}_{\varvec{T}} \varvec{U}_{\varvec{T}}^\mathsf {T}) \otimes (\varvec{U}_{\varvec{A}}\varvec{U}_{\varvec{A}}^\mathsf {T}) = (\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})(\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})^\mathsf {T}\). Rearranging the above equation, we get

figure a
$$\begin{aligned} \varvec{C} = (\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})(\varvec{\varLambda }_{\varvec{T}} \otimes \varvec{\varLambda }_{\varvec{A}}+\sigma ^2I_{n\times n})(\varvec{U}_{\varvec{T}} \otimes \varvec{U}_{\varvec{A}})^\mathsf {T}, \end{aligned}$$

from which decomposition of \(\varvec{C}^{-1}\) in Eq. (15) follows directly.

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Caponera, A., Denti, F., Rigon, T., Sottosanti, A., Gelfand, A. (2018). Hierarchical Spatio-Temporal Modeling of Resting State fMRI Data. In: Canale, A., Durante, D., Paci, L., Scarpa, B. (eds) Studies in Neural Data Science. START UP RESEARCH 2017. Springer Proceedings in Mathematics & Statistics, vol 257. Springer, Cham. https://doi.org/10.1007/978-3-030-00039-4_7

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