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Bayesian Estimation of Conditional Independence Graphs Improves Functional Connectivity Estimates.

Hinne M, Janssen RJ, Heskes T, van Gerven MA - PLoS Comput. Biol. (2015)

Bottom Line: A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections.As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results.The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

View Article: PubMed Central - PubMed

Affiliation: Radboud University, Institute for Computing and Information Sciences, Nijmegen, the Netherlands.

ABSTRACT
Functional connectivity concerns the correlated activity between neuronal populations in spatially segregated regions of the brain, which may be studied using functional magnetic resonance imaging (fMRI). This coupled activity is conveniently expressed using covariance, but this measure fails to distinguish between direct and indirect effects. A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections. Importantly, provided the data are normally distributed, if two variables are conditionally independent given all other variables, their respective partial correlation is zero. In this paper, we propose a probabilistic generative model that allows us to estimate functional connectivity in terms of both partial correlations and a graph representing conditional independencies. Simulation results show that this methodology is able to outperform the graphical LASSO, which is the de facto standard for estimating partial correlations. Furthermore, we apply the model to estimate functional connectivity for twenty subjects using resting-state fMRI data. Results show that our model provides a richer representation of functional connectivity as compared to considering partial correlations alone. Finally, we demonstrate how our approach can be extended in several ways, for instance to achieve data fusion by informing the conditional independence graph with data from probabilistic tractography. As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results. This reveals that while we are able to infer a clear backbone of connectivity in our empirical results, the data are not accurately described by simply looking at the mode of the distribution over connectivity. The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

No MeSH data available.


The histograms for each of the 28 different simulations.Positive error z-scores indicate that the point estimate was less effective in recovering the ground truth than the Gaussian graphical model, while the reverse is true for negative error z-scores. The red dashed lines indicate the interval outside of which the difference in performance is significant (p < 0.01, z-test). Note the different ordinate axes.
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pcbi.1004534.g003: The histograms for each of the 28 different simulations.Positive error z-scores indicate that the point estimate was less effective in recovering the ground truth than the Gaussian graphical model, while the reverse is true for negative error z-scores. The red dashed lines indicate the interval outside of which the difference in performance is significant (p < 0.01, z-test). Note the different ordinate axes.

Mentions: Fig 3 shows the (smoothed) histograms of z-scores aggregated over the 50 runs per simulation, for the graphical LASSO approach with λ = 100 (the results for λ = 5 and the MLE are almost identical; the MLE results are shown in S1 Fig). In this figure, distributions of errors with high z-scores have substantially larger errors than the errors from the BGGM approach, while distributions with low z-scores have smaller errors. The significance threshold at p < 0.01 is indicated by the red dotted lines. The first row of Fig 3 shows the total scores (both true positives and true negatives) for each simulation, while the second and the third row split this score into the contributions for true positive connections and true negative connections, respectively. These results indicate that in terms of true positives, the LASSO approach typically has an equal to slightly better performance than our Bayesian alternative. However, the BGGM approach identifies true negatives at least as well as G-LASSO, and in several cases significantly outperforms it. On the whole, the proposed method is up to par with the graphical LASSO (for λ ∈ {5, 100}) and the MLE, while at times outperforming them greatly.


Bayesian Estimation of Conditional Independence Graphs Improves Functional Connectivity Estimates.

Hinne M, Janssen RJ, Heskes T, van Gerven MA - PLoS Comput. Biol. (2015)

The histograms for each of the 28 different simulations.Positive error z-scores indicate that the point estimate was less effective in recovering the ground truth than the Gaussian graphical model, while the reverse is true for negative error z-scores. The red dashed lines indicate the interval outside of which the difference in performance is significant (p < 0.01, z-test). Note the different ordinate axes.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4634993&req=5

pcbi.1004534.g003: The histograms for each of the 28 different simulations.Positive error z-scores indicate that the point estimate was less effective in recovering the ground truth than the Gaussian graphical model, while the reverse is true for negative error z-scores. The red dashed lines indicate the interval outside of which the difference in performance is significant (p < 0.01, z-test). Note the different ordinate axes.
Mentions: Fig 3 shows the (smoothed) histograms of z-scores aggregated over the 50 runs per simulation, for the graphical LASSO approach with λ = 100 (the results for λ = 5 and the MLE are almost identical; the MLE results are shown in S1 Fig). In this figure, distributions of errors with high z-scores have substantially larger errors than the errors from the BGGM approach, while distributions with low z-scores have smaller errors. The significance threshold at p < 0.01 is indicated by the red dotted lines. The first row of Fig 3 shows the total scores (both true positives and true negatives) for each simulation, while the second and the third row split this score into the contributions for true positive connections and true negative connections, respectively. These results indicate that in terms of true positives, the LASSO approach typically has an equal to slightly better performance than our Bayesian alternative. However, the BGGM approach identifies true negatives at least as well as G-LASSO, and in several cases significantly outperforms it. On the whole, the proposed method is up to par with the graphical LASSO (for λ ∈ {5, 100}) and the MLE, while at times outperforming them greatly.

Bottom Line: A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections.As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results.The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

View Article: PubMed Central - PubMed

Affiliation: Radboud University, Institute for Computing and Information Sciences, Nijmegen, the Netherlands.

ABSTRACT
Functional connectivity concerns the correlated activity between neuronal populations in spatially segregated regions of the brain, which may be studied using functional magnetic resonance imaging (fMRI). This coupled activity is conveniently expressed using covariance, but this measure fails to distinguish between direct and indirect effects. A popular alternative that addresses this issue is partial correlation, which regresses out the signal of potentially confounding variables, resulting in a measure that reveals only direct connections. Importantly, provided the data are normally distributed, if two variables are conditionally independent given all other variables, their respective partial correlation is zero. In this paper, we propose a probabilistic generative model that allows us to estimate functional connectivity in terms of both partial correlations and a graph representing conditional independencies. Simulation results show that this methodology is able to outperform the graphical LASSO, which is the de facto standard for estimating partial correlations. Furthermore, we apply the model to estimate functional connectivity for twenty subjects using resting-state fMRI data. Results show that our model provides a richer representation of functional connectivity as compared to considering partial correlations alone. Finally, we demonstrate how our approach can be extended in several ways, for instance to achieve data fusion by informing the conditional independence graph with data from probabilistic tractography. As our Bayesian formulation of functional connectivity provides access to the posterior distribution instead of only to point estimates, we are able to quantify the uncertainty associated with our results. This reveals that while we are able to infer a clear backbone of connectivity in our empirical results, the data are not accurately described by simply looking at the mode of the distribution over connectivity. The implication of this is that deterministic alternatives may misjudge connectivity results by drawing conclusions from noisy and limited data.

No MeSH data available.