[Paper Review] Bayesian estimation for large scale multivariate Ornstein-Uhlenbeck model of brain connectivity

Andrea Insabato, John P. Cunningham+1 more|arXiv (Cornell University)|May 25, 2018

Functional Brain Connectivity Studies12 references3 citations

TL;DR

This paper compares Bayesian estimation with moments and Lyapunov optimization for large-scale multivariate Ornstein-Uhlenbeck (mOU) models of brain connectivity. It shows that Bayesian estimation is equivalent to the method of moments but suffers from poor scalability and numerical instability—especially due to growing imaginary components in connectivity estimates—making it significantly less accurate than Lyapunov optimization, which achieves better classification performance on real fMRI data with fewer time samples.

ABSTRACT

Estimation of reliable whole-brain connectivity is a crucial step towards the use of connectivity information in quantitative approaches to the study of neuropsychiatric disorders. When estimating brain connectivity a challenge is imposed by the paucity of time samples and the large dimensionality of the measurements. Bayesian estimation methods for network models offer a number of advantages in this context but are not commonly employed. Here we compare three different estimation methods for the multivariate Ornstein-Uhlenbeck model, that has recently gained some popularity for characterizing whole-brain connectivity. We first show that a Bayesian estimation of model parameters assuming uniform priors is equivalent to an application of the method of moments. Then, using synthetic data, we show that the Bayesian estimate scales poorly with number of nodes in the network as compared to an iterative Lyapunov optimization. In particular when the network size is in the order of that used for whole-brain studies (about 100 nodes) the Bayesian method needs about eight times more time samples than Lyapunov method in order to achieve similar estimation accuracy. We also show that the higher estimation accuracy of Lyapunov method is reflected in a much better classification of individuals based on the estimated connectivity from a real dataset of BOLD fMRI. Finally we show that the poor accuracy of Bayesian method is due to numerical errors, when the imaginary part of the connectivity estimate gets large compared to its real part.

Motivation & Objective

• Address the challenge of estimating reliable whole-brain connectivity from limited fMRI time series with high dimensionality.
• Evaluate the performance of Bayesian estimation in comparison to moments and Lyapunov optimization for large-scale mOU models.
• Investigate the causes of poor estimation accuracy in Bayesian methods, particularly numerical instability.
• Assess the impact of estimation accuracy on downstream classification of individuals using estimated connectivity.
• Explore the potential for improving Bayesian estimation through alternative priors or regularization techniques.

Proposed method

• Formalize the multivariate Ornstein-Uhlenbeck (mOU) process as a network model with coupling matrix $C$, noise variance $\Sigma$, and time constant $\tau_x$.
• Use the Lyapunov equation $JQ^0 + Q^0J^T + \Sigma = 0$ to compute the theoretical covariance $Q^0$ from model parameters.
• Apply the moments method by estimating $\hat{J}$ via $\hat{J} = \frac{1}{\tau} \left[ \log m(\hat{Q}^\tau (\hat{Q}^0)^{-1}) \right]^T$, where $\log m$ is the matrix logarithm.
• Implement Bayesian estimation using uniform priors, equivalent to the method of moments, and compute posterior mean estimates.
• Use iterative Lyapunov optimization to refine estimates by minimizing error in the covariance and lagged-covariance matrices.
• Apply structural connectivity as a mask to constrain connectivity estimates and improve interpretability and sparsity.

Experimental results

Research questions

• RQ1 Is Bayesian estimation for the mOU model equivalent to the method of moments when using uniform priors?
• RQ2 How does the estimation accuracy of Bayesian methods scale with network size and number of time samples compared to Lyapunov optimization?
• RQ3 What is the primary source of estimation error in Bayesian mOU estimation, particularly in large-scale brain networks?
• RQ4 Does improved estimation accuracy from Lyapunov optimization translate into better performance in downstream classification tasks on real fMRI data?
• RQ5 Can numerical instability in the matrix logarithm step be attributed to the growing ratio of imaginary to real parts in the connectivity estimate?

Key findings

• Bayesian estimation with uniform priors is mathematically equivalent to the method of moments for mOU model parameter estimation.
• The Bayesian method exhibits significantly worse estimation accuracy than Lyapunov optimization, requiring approximately eight times more time samples to achieve similar performance on 100-node networks.
• The primary source of error in Bayesian estimation is numerical instability in the matrix logarithm step, where the imaginary part of the estimated connectivity grows disproportionately relative to its real part as network size increases.
• The Lyapunov method achieves higher classification accuracy for subject identity using real fMRI data, with violin plot distributions showing consistently better performance across 100 random train-test splits.
• The estimation error in Bayesian methods is not due to poor covariance or lagged-covariance estimation, but specifically due to the matrix logarithm operation on the ratio $\hat{\Lambda} = (\hat{Q}^0)^{-1} \hat{Q}^\tau$.
• The ratio of the norm of the imaginary part to the real part of the estimated connectivity increases with network size, confirming that numerical instability in the logarithm is the dominant error source.