connections weights). An interesting observation is that although the exact covariance ma- trix Σ is ill-conditioned (remember that its condition number is equal to 401) and there is only a very small number of samples available, optimization (2.3) detects the right structure of the circuit. In the next section, the graphical lasso algorithm and its modified version will be applied to fMRI data for comparison.
2.4 FMRI data: Graphical Lasso vs. Modified Graphical Lasso
Consider the data set available in [44] in which resting state fMRI data was acquired for a group of 20 healthy subjects. 134 samples of the low frequency neurophysiological os- cillations were taken at 140 cortical brain regions in the right hemisphere. The 140×140 sample covariance matrix Σs can be computed for each subject from this data set. Note that the number of samples is smaller than the number of variables, and therefore Σs is ill-conditioned and non-invertible. Figure 2.5 shows the 2-D picture of the 140 nodes (brain regions).
The aim of this section is to model the brain connectivity network using the graphical
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Figure 2.5: 2-D picture of the 140 brain regions.
lasso algorithm (2.1) and the modified graphical lasso (2.3). For both of these two opti- mization problems, the regularization parameter α is to be chosen in such a way that the resulting graph will have the sparsest possible structure and yet be a connected graph. We will solve optimizations (2.1) and (2.3) for resting-state fMRI data of three different sub- jects, called Subject 1, Subject 2 and Subject 3. The connectivity (concentration) graph of each subject will then be plotted as follows:
• The strong connections are plotted in black (we consider a connection strong if its weight is at least 10 times larger than average of the absolute values of all weights).
Furthermore, the width of each black line in the graph represents the strength of the connection, which means stronger connections are shown with thicker lines.
• The rest of the edges (weak connections) are shown in gray.
As mentioned before, the graphs of the inverse correlation matrix (R−1) and the inverse covariance matrix (Σ−1) have the same sparsity pattern. Therefore, one may feed the sample correlation matrix Σs instead of the sample covariance matrix into the graphical lasso algorithm (2.1) and the modified optimization (2.3). Simulations on the fMRI data show that the graphs based on the sample correlation matrix are much sparser (by a factor of 2) than the ones based on the sample covariance matrix. Therefore, we substitute the sample covariance matrix Σs in optimizations (2.1) and (2.3) with the sample correlation matrix of the resting state fMRI data of each subject. Note that each of graphs obtained from these algorithms is an estimatedconcentration graph.
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Figure 2.6: (a) The sparsest connected graph for the resting-state-fMRI data of Subject 1 obtained from optimization (2.1) for α = 0.315. (b) The sparseness of the off-diagonal entries of the solution of optimization (2.1) for Subject 1 after taking the absolute value of its elements.
I) Graphical lasso and brain connectivity
Subject 1: For the resting-state fMRI data acquired from Subject 1, the sparsest con- nected graph that can be obtained from optimization (2.1) is depicted in Figure 2.6(a).
This graph, corresponding toα= 0.315, has 987 edges connecting the 140 spatially disjoint brain regions. The color map depicted in Figure 2.6(b) illustrates the sparseness of the off-diagonal entries of the matrix solution obtained from optimization (2.1) after taking the absolute value of the matrix elements.
Subject 2: For the resting-state fMRI data acquired from Subject 2, the sparsest con- nected graph that can be obtained from optimization (2.1) is depicted in Figure 2.7(a). This
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Figure 2.7: (a) The sparsest connected graph for the resting-state-fMRI data of Subject 2 obtained from optimization (2.1) for α = 0.355. (b) The sparseness of the off-diagonal entries of the solution of optimization (2.1) for Subject 2 after taking the absolute value of its elements.
graph, corresponding toα = 0.355, has 764 edges. Figure 2.7(b) illustrates the sparseness of off-diagonal entries of the solution of optimization (2.1) after taking the absolute value of the matrix elements.
Subject 3: For the resting-state fMRI data of Subject 3, the sparsest connected graph that can be obtained from optimization (2.1) is depicted in Figure 2.8(a). This graph, corresponding to α = 0.275, has 998 edges. Figure 2.8(b) illustrates the sparseness of off- diagonal entries of the solution of optimization (2.1) after taking the absolute value of the matrix elements.
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Figure 2.8: (a) The sparsest connected graph for the resting-state-fMRI data of Subject 3 obtained from optimization (2.1) for α = 0.275. (b) The sparseness of the off-diagonal entries of the solution of optimization (2.1) for Subject 3 after taking the absolute value of its elements.
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Figure 2.9: (a) The sparsest connected graph for the resting-state-fMRI data of Subject 1 obtained from optimization (2.3) for α = 0.445 and β = 5. (b) The sparseness of the off- diagonal entries of the solution of optimization (2.3) for Subject 1 after taking the absolute value of its elements.
II) Modified graphical lasso and brain connectivity
Subject 1: For the resting-state fMRI data acquired from Subject 1, the sparsest con- nected graph that can be obtained from optimization (2.3) is depicted in Figure 2.9(a).
This graph, obtained for α = 0.315 and β = 5, has 608 edges. This graph has 595 edges in common with the graph depicted in Figure 2.6(a) for the same subject but obtained by solving the graphical lasso algorithm (2.1). Figure 2.9(b) illustrates the sparseness of the off-diagonal entries of the solution of optimization (2.3) after taking the absolute value of its elements.
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Figure 2.10: (a) The sparsest connected graph for the resting-state-fMRI data of Subject 2 obtained from optimization (2.3) for α = 0.356 and β = 5. (b) The sparseness of the off- diagonal entries of the solution of optimization (2.3) for Subject 2 after taking the absolute value of its elements.
Subject 2: For the resting-state fMRI data acquired from Subject 2, the sparsest con- nected graph that can be obtained from optimization (2.3) is depicted in Figure 2.10(a).
This graph, obtained for α = 0.356 and β = 5, has 540 edges and is a subset of the graph depicted in Figure 2.7(a) for the same subject but obtained by solving the graphical lasso algorithm (2.1). Figure 2.10(b) illustrates the sparseness of the off-diagonal entries of the solution of optimization (2.3) after taking the absolute value of its elements.
Subject 3: For the resting-state fMRI data acquired from Subject 3, the sparsest con- nected graph that can be obtained from optimization (2.3) is depicted in Figure 2.11(a).
This graph, obtained for α = 0.275 and β = 5, has 688 edges out of which 680 edges are
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Figure 2.11: (a) The sparsest connected graph for the resting-state-fMRI data of Subject 3 obtained from optimization (2.3) for α = 0.275 and β = 5. (b) The sparseness of the off- diagonal entries of the solution of optimization (2.3) for Subject 3 after taking the absolute value of its elements.
in common with the graph depicted in Figure 2.8(a) for the same subject but obtained by solving the graphical lasso algorithm (2.1). Figure 2.11(b) illustrates the sparseness of the off-diagonal entries of the solution of optimization (2.3) after taking the absolute value of its elements.
The above simulations are summarized in Table 2.1. In summary, the graph obtained from the modified graphical lasso for each subject is not only sparser than but also mostly a subgraph of the one obtained from the graphical lasso. It can be verified that the graphs of Subjects 1-3 obtained from the modified graphical lasso have 62 common edges. The common subgraph of these three graphs is shown in Figure 2.12.
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Figure 2.12: The 62 edges that are in common among the graphs of Subjects 1-3 obtained from optimization (2.3).
Subject Optimization α β Edges
1 (2.1) 0.315 0 987
1 (2.3) 0.315 5 608
2 (2.1) 0.355 0 764
2 (2.3) 0.356 5 540
3 (2.1) 0.275 0 998
3 (2.3) 0.275 5 688
Table 2.1: This table shows the number of edges for the graphs obtained from optimization (2.1) and optimization (2.3) for Subjects 1-3.