2.3.1 Motivation for new method.

Although OCF has been demonstrated to be successful in decomposing the variability of connectivity matrices [18], we noticed some of its drawbacks that might prevent it from being widely applicable. The following issues particularly motivated us to develop the new MCF method which we describe below.

First, OCF implicitly assumes that the variability of connectivity within each module (i.e., intra-module connectivity) is almost negligible, where the two self-product terms, and , are dropped in Eq (3). If this assumption does not hold, the two weight vectors necessarily overlap to spuriously produce nonzero patterns within individual modules (i.e., diagonal blocks of eigenconnectivity matrix; see Fig 1(b)). This inseparability of intra- and inter-module connectivities is problematic because it complicates the interpretation of weight vectors and might even distort the inter-module (off-diagonal) parts of the eigenconnectivity if the intra-module parts need to have large values.

Second, the weight and eigenconnectivity patterns obtained by OCF might contain both positive and negative signs in one module (Fig 1(b)), which is often difficult to interpret. The different signs in eigenconnectivity mean that the connectivities corresponding to positive and negative signs vary in opposite directions. If one increases, the other decreases, and vice versa. Hence, mixed eigenconnectivity signs imply that a module may be divided into submodules with distinct functional properties, characterized by strong anti-correlations (i.e., negative correlations) in the variability of their connectivities to any other modules as well as the variability in their internal connectivities, if intra-module connectivity is also taken into account. Such inhomogeneity would be more interpretable if it were represented explicitly as different modules.

Finally, OCF focuses on the variability of connectivity between a pair of modules, and thus it is not applicable to finding general eigenconnectivity patterns among more than two modules. Although an empirical observation [18] strongly motivated the rank-two approximation, it does not necessarily imply that the information discarded by the approximation is meaningless. In particular, we can expect to obtain eigenconnectivity patterns that are more interesting from any neurophysiological (or any other domain-specific) perspective by increasing the rank and thus the number of modules.

2.3.2 MCF’s representation of eigenconnectivity matrix.

We introduce a new parametric representation of simplified eigenconnectivity matrix B by extending the one of OCF Eq (3). To begin with, we only consider a pair of modules. The general case of more than two modules is described in Section 2.3.5.

The new parametric form of B is given as follows, which is also illustrated in Fig 1(c): (5) where w_k is the weight vector for module k, such that node j belongs to module k when its j-th element w_jk is nonzero. The two weight vectors are collected in the D × 2 matrix W = (w₁, w₂). Real coefficients g_kl determine the relative magnitudes and the signs of these outer products and are collected in a 2 × 2 matrix G = (g_kl). For simplicity, we assume that matrix G is non-singular, so that the number of modules equals the rank of B. To avoid an obvious ambiguity in the scaling between W and G, we fix every w_k to have unit ℓ₂-norm, i.e., ‖w_k‖ = 1 for every k.

Next we explain additional assumptions on parameters W and G in detail:

1. Variabilities in both intra- and inter-module connectivities: As a key difference from the parametrization Eq (3) of OCF, we let every diagonal element g_kk of G be generally nonzero, so that the new one Eq (5) represents both the intra-module and inter-module connectivities with outer products and (k ≠ l). Coefficient g_kl specifies the overall strengths and signs of the corresponding term in Eq (5), since the outer products are normalized commonly as . In what follows, we refer to G as a module-level eigenconnectivity matrix, representing node-level B in a compact form.
2. Disjointness of modules: To avoid the spurious effects on intra-module eigenconnectivity due to the overlap of weight vectors among modules, we further constrain the weight vectors so that their nonzero elements do not overlap among modules. That is, we set W ∈ Ω, where (6) Then the modules are mutually disjoint, and each and (k ≠ l) represent clearly the intra- and inter-module eigenconnectivity patterns. Note that the weight vectors are now mutually orthonormal (i.e., W^⊤ W = I) as in OCF, which readily results from W ∈ Ω and the unit-norm constraints on the columns as introduced above.
3. Nonnegativity/Uniform signs: To improve the interpretability of the obtained weight vectors, we further constrain each module’s weights to have uniform signs. Notice that the sign of each module is actually arbitrary and is eventually absorbed into G. Here, we thus specifically set every weight to be nonnegative, i.e., (7) where denotes the set of nonnegative real numbers.

The matrix factorization Eq (5) with these specific assumptions as above still has some indeterminacy in parameters W and G. An obvious indeterminacy is the permutation of the modules. That is, relation B = WGW^⊤ holds even if W and G are replaced with WP and P^⊤ GP for any permutation matrix P without violating the above constraints on W. Another indeterminacy is that the global sign of G can always be flipped since the global sign of B is immaterial, as in PCA. In practice, these indeterminacies do not seem to cause serious problems, since the permutation and the global sign may be arbitrarily chosen after learning so that the interpretation is easier.

2.3.3 Stepwise MCF via post-hoc matrix factorization.

Before presenting the principled constrained PCA approach based on the new eigenconnectivity representation, we first introduce a stepwise approach (Stepwise MCF) to approximately obtain W and G based on the eigenconnectivity matrix B_PCA precomputed by PCA. The stepwise approach is useful in its own right, while we specifically use it to effectively initialize the iterative algorithm of constrained PCA (see Section 2.3.4). In the following, we briefly sketch the algorithm; the details are found in S1 Appendix (Section A).

Algorithm 1 summarizes the entire procedure of Stepwise MCF to obtain a single MCF component, which essentially consists of the following steps: first, 1) we solve the unconstrained PCA problem Eq (2) to obtain eigenconnectivity matrix B_PCA, and then 2) compute the two eigenvectors of B_PCA corresponding to the largest eigenvalues (in absolute value) to form a D × 2 matrix U. Finally, we 3) alternatingly optimize an orthogonal matrix V and W ∈ Ω₊ so that the error between UV^⊤ and W is minimized, where (8) This can be solved by a simple modification of a technique previously developed for (multiclass) spectral clustering [19], and an additional column-wise normalization further ensures W to satisfy all of the original constraints, i.e., W ∈ Ω₊ and ∀k‖w_k‖ = 1. We further 4) set G = W^⊤ B_PCA W to obtain an approximate factorization B_PCA ≈ WGW^⊤.

In this procedure, we randomly initialize V as an orthogonal matrix and flip the signs of its columns to make every entry of nonnegative, where and 1 denotes an all-one vector. This simple heuristics ensures that the columns of UV^⊤ to be summed up are nonnegative, so that UV^⊤ becomes closer to nonnegative W. The projection to W ∈ Ω₊ possibly results in zero column vectors in W due to the lack of explicit norm constraint. To avoid this, we detect zero columns to restart the procedure from re-initialized V.

Algorithm 1 Stepwise MCF (single component)

Require: Connectivity matrices {X_n}; constant ϵ > 0

1: Perform PCA on connectivity matrices to obtain B_PCA

2: Compute the best rank-two eigenvalue decomposition B_PCA ≈ UQU^⊤;

3: Randomly generate orthogonal matrix V

4:

5: repeat

6:  V_old ← V

7:       ▹ Projection to the set Ω₊ (see S1 Appendix)

8:  if any column in W is zero vector, then

9:   Randomly generate orthogonal matrix V

10:   

11:  else

12:   Compute singular value decomposition: U^⊤ W = L Σ R^⊤; V ← RL^⊤

13:  end if

14: until

15: Normalize every column in W to have unit ℓ₂-norm

16: G ← W^⊤ B_PCA W

We remark that the approximate factorization B_PCA ≈ WGW^⊤ is in fact applicable to any matrix B other than that obtained by PCA. Thus, Algorithm 1 (except for the first line) can readily be used, e.g., to analyze a single connectivity matrix X_n or the average ; the stepwise analysis can even be combined with any methods other than PCA, such as k-means clustering (see S2 Appendix) and ICA.

2.3.4 Constrained PCA algorithm.

Plugging the factorized eigenconnectivity matrix Eq (5) into the original PCA problem Eq (2), we have the following constrained PCA optimization problem to obtain a single MCF component: (9) The additional unit-norm constraint on G ensures ‖B‖ = 1, since W^⊤ W = I implies ‖WGW^⊤‖ = ‖G‖. Note that optimal G is symmetric because every is symmetric; if we further constrain G so that every diagonal entry is zero, constant in OCF Eq (3) emerges in the off-diagonal entries under the unit-norm constraint on G. An alternative interpretation of Problem (9) is given in S1 Appendix (Section B).

To solve Eq (9), we use a simple alternating maximization scheme with auxiliary variables, similar to [18]. Let r = (r₁, r₂, …, r_N)^⊤ be an auxiliary vector variable satisfying ‖r‖ = 1, and define . Then the square root of the objective function in Eq (9) has the following lower bound: (10) where equality holds when r_n equals times a constant factor to ensure ‖r‖ = 1. Since taking the square root does not change the solution of the Problem (9), the Problem (9) is equivalent to the maximization of the right-hand side in Eq (10). The lower bound is then maximized alternatingly between (W, G) and r.

The solution for r was given above. To derive the solution for (W, G), we first analytically solve for G, yielding G = W^⊤ CW/‖W^⊤ CW‖, and then substitute it into the right-hand side of Eq (10). We again square the objective without changing the optimal solution and eventually have a reduced problem only for W: (11) which we solve with a gradient projection technique (see, e.g., [20]). Note that we do not need to perfectly solve this inner maximization Eq (11), but we only need to increase the objective. In practice, we perform the gradient projection step only once at each alternating maximization step.

We introduce a single gradient projection step as the following update of W: (12) where matrix ΔW defines the search direction in the space of W, η ≥ 0 is a stepsize parameter, and is the projection operator to keep W in the feasible set (see below). Denote by f(W) the objective function in Eq (11) and by ∇_W f(W) its gradient, derived in S1 Appendix (Section C) as (13) Then, according to a theory of optimization on manifold [21], we set the search direction as (14) which restricts the gradient in the tangent space of the Stiefel manifold (i.e., the set of Ws such that W^⊤ W = I) at W. We choose the stepsize η at each update by a backtracking line search to satisfy Armijo’s condition (e.g., [20]).

The projection operator in Eq (12) still needs to be specified. Unfortunately, exact orthogonal projection of W to feasible set {W∣W ∈ Ω₊, ∀k‖w_k‖ = 1} has no simple solution. Thus we use an approximate method and first project W to Ω₊ without norm constraints, followed by the normalization of every column so that each has a unit norm. Note that the first step can be solved by Proposition 1, given in S1 Appendix (Section A), and the subsequent normalization does not violate W ∈ Ω₊, so that W eventually belongs to the feasible set. Algorithm 2 summarizes the overall algorithm.

Algorithm 2 MCF (single component): solution to constrained PCA Eq (9)

Require: N connectivity matrices ; initial stepsize η > 0; α ∈ (0, 1) for Armijo condition, β ∈ (0, 1) for backtracking line search, and ϵ > 0 for stopping the criterion

1: Initialize W and G using Algorithm 1

2: repeat

3:   for every n

4:  Normalize r to have unit ℓ₂-norm

5:  

6:  ∇f(W) ← 4CWW^⊤ CW

7:  ΔW ← ∇f(W) − W(∇f(W))^⊤ W

8:  W_old ← W

9:  repeat

10:        ▹ Projection to the set Ω₊ (see S1 Appendix)

11:   Normalize each column of W′ to have unit ℓ₂-norm

12:   η ← βη

13:  until f(W′) ≤ f(W) + αtr[(∇f(W))^⊤(W′ − W)]

14:  W ← W′

15: until ‖W^⊤ W_old − I‖ < ϵ

16: Compute G = W^⊤ CW and normalize G to have unit Frobenius norm

2.3.5 Generalization to K modules.

So far we have only considered two modules to represent a single eigenconnectivity matrix, B. In fact, the current formulation allows the number of modules to be any positive integer smaller than D, which further generalizes MCF. Let K denote the number of modules. Then weight matrix W has K columns, and module-level eigenconnectivity G is a K × K square matrix. The above algorithms remain valid, except that the rank-2 approximation in line 2 of Algorithm 1 must be replaced with rank-K approximation.

As an exploratory analysis method, however, we must usually avoid setting the numbers of modules too large. The standard choice of K = 2 is thus reasonable since it gives the minimal case that includes both intra- and inter-module connectivities. We can also increase K so that the patterns obtained are more meaningful in terms of domain knowledge, although the module-level eigenconnectivity patterns might quickly become too complicated. We usually set K to less than about five so that the result can be easily visualized.

To support the selection of reasonable numbers of modules K, we can also examine the eigenvalue spectrum of B_PCA: Let q₁, q₂, …, q_D be the eigenvalues of B_PCA, sorted in nonincreasing order. Then the power (squared Frobenius norm) of B_PCA is orthogonally decomposed into squared eigenvalues . Given the spectrum, we can use any conventional heuristics in the PCA literature to determine which eigenvalues’ contributions are meaningful (see, e.g., [22]). In practice, such an approach can be easily combined with Stepwise MCF (Algorithm 1). In the subsequent optimization of the constrained PCA (Algorithm 2), we simply use the same K selected at the initialization stage.

2.3.6 Extraction of multiple components.

The solution to Eq (9) gives the first MCF component in terms of W and G. Similar to PCA, a deflation method can be used to obtain subsequent components, solving Eq (9) in a recursive manner. Given corresponding components , we replace every instance with the residual unexplained by the preceding components: (15) and then the next component’s W and G can be obtained by solving Eq (9) again with updated .

According to this deflation scheme, the m-th step yields an approximate decomposition of centered connectivity matrices in terms of basis {B₁, B₂, …, B_m} as well as the corresponding components denoted by s_mn. Note that matrices B₁, B₂, …, B_m are not necessarily orthogonal to each other (i.e., dot-products are not necessarily zero), so that the total variance (in the data space) explained by the m components cannot be simply computed by summing up the variances of components s_mn as in a standard way.

Here, following the idea of adjusted total variance [15], which was originally developed for sparse PCA, we evaluate the total explained variance by MCF components with a correction to the non-orthogonality. First we orthogonalize the basis using the well-known Gram-Schmidt method to get with orthonormal basis and adjusted components . Then we compute the adjusted total variance explained by the MCF components by summing the adjusted component variances, i.e., .

2.3.7 Relation to nonnegative tensor factorization.

Interestingly, our MCF developed above turns out to be closely related to nonnegative tensor factorization (see, e.g., [23]) which has recently also found applications in the brain’s functional network analyses [24]; the goal is rather different from eigenconnectivity analysis. In S3 Appendix, we compare our method with a particular type of nonnegative tensor factorization as well and discuss their fundamental differences.

2.4.1 Simulation I: illustrative example.

First, we performed a simple simulation to illustrate how our new MCF method improves the existing methods for eigenconnectivity analysis, namely, PCA and OCF. In particular, we examined the effect of increasing the variability of intra-module connectivity on the estimated eigenconnectivity and the weight matrix patterns.

Here, connectivity matrices X_n were created simply by multiplying a random number generated from standard Gaussian distribution to a true eigenconnectivity matrix B and adding symmetric random noise E_n. The number of nodes was set to D = 20. True B was created as B = WGW^⊤, where two unit-norm columns w₁ and w₂ were given so that only entries w_j1 for j ∈ {4, 5, …, 8} and w_j2 for j ∈ {12, 13, …, 18} were nonzero. These nonzero values were set uniform in each column. Noise E_n was generated by first sampling the upper triangular part as random Gaussian white noise (SD: 0.3) and copying it into the strictly lower triangular part to form a symmetric matrix. Matrix G was specifically given as (16) for some values of c ∈ [0, 1]. Notice that the sum of the squares in the diagonal and off-diagonal elements satisfy and . Thus parameter c gives the relative strength of the intra-module network variability in the data.

We applied PCA, OCF and MCF (Algorithm 2, K = 2) to the collection of connectivity matrices synthesized as above. We used a relatively large sample size, N = 10,000, to clarify the difference particularly due to the modeling assumptions in those methods rather than stochastic estimation errors. We implemented all the methods in Matlab; we used eigs function for PCA and the original code for OCF (https://www.cs.helsinki.fi/u/ahyvarin/code/ocf/). We set the tolerance parameters ϵ reasonably small, i.e., ϵ = 10⁻¹² in Algorithm 1 and ϵ = 10⁻⁶ in Algorithm 2. In Algorithm 2, we also set the initial stepsize as η = 0.01, Armijo parameter α = 10⁻⁴ as suggested in [25], and β = 0.5 to halven the stepsize in each backtracking step. These settings were common for all the experiments below.

2.4.2 Simulation II: quantitative comparison.

Next, we performed a more quantitative evaluation of the estimation errors by the above methods. Here, we randomly synthesized N symmetric matrices X₁, X₂, …, X_N of size 100 × 100 (D = 100) according to the following generative model, (17) so that unit-norm symmetric matrices B₁ and B₂ define a two-dimensional subspace in the matrix space, along which observed matrices X_n largely varied. The coefficients, s_1n and s_2n, were randomly generated by zero-mean Gaussian distributions with the standard deviations (SDs) given by 1 and 0.6. Random noise E_n was again generated from Gaussian white noise (SD: 0.3), symmetrized appropriately as above.

Matrices B₁ and B₂ were created randomly in each run of the simulation in the following manner. First, we generated a 100 × 10 weight matrix W by randomly partitioning the D nodes into ten non-singleton subsets (this step simply produced modules of moderate sizes; only the four columns in W were selected and the remaining six were discarded). and sampling the corresponding nonzero weights w_jk from uniform distribution over [0.5, 1.5], followed by renormalizing each column of W to have unit ℓ₂-norm. Then we randomly chose two columns in W to form W₁ and another two columns from the remaining eight columns of W to form W₂. Finally, we generated 2 × 2 symmetric matrices G₁ and G₂ from the Gaussian distribution, renormalized them to have a unit Frobenius norm, and set for m = 1, 2. To generate each G, we either 1) explicitly set the diagonal elements at zero (i.e., g_kk = 0 for any k, as OCF assumes) before normalization or 2) randomly sampled those diagonal elements as well.

We compared 1) the standard PCA on connectivity matrices, 2) OCF, 3) Stepwise MCF (Algorithm 1), which was also used to initialize MCF, and 4) MCF, using the projected gradient method (Algorithm 2). We ran these four algorithms in the two different conditions on true G explained above.

By construction, B₁ and B₂ synthesized above were mutually orthogonal and thus serve as the orthonormal basis of the two-dimensional principal subspace. B₁ and B₂ gave the true eigenconnectivity patterns of the first and second PCs to each of which corresponding estimate should approach. Thus, as a basic performance measure, we evaluated estimation error for m = 1, 2, i.e., the root mean squared error (RMSE) between the corresponding elements of the two matrices. We ran the simulation 100 times for given sample size N and evaluated the estimation error in each of the 100 runs.

2.5.1 Dataset.

The dataset consisted of a total of 986 functional connectivity matrices, which we obtained from the USC Multimodal Connectivity Database (http://umcd.humanconnectomeproject.org) [26]. We used the data tagged “1000_ Functional_ Connectomes” and removed 17 connectivity matrices from the original 1003 because the files seemed corrupt. Each connectivity matrix corresponded to one subject, consisting of pairwise correlations of blood-oxygen-level dependent (BOLD) signals between 177 functional regions-of-interest (ROIs) that were obtained using the spatially constrained spectral clustering method [27].

2.5.2 Data analysis.

We applied both OCF and MCF (K = 2) to this dataset and extracted two components with each method. Due to the existence of local optima in the objectives, we used 20 different initial conditions and examined the results that achieved the maximum explained variance. In fact, the result was qualitatively not sensitive to initial conditions; MCF runs deterministically once initialized by the output of Algorithm 1 and we empirically found that Algorithm 1 was quite robust to the initial choice of V.

We also demonstrated the generalization of MCF to more than two modules (Section 2.3.5) using the same fcMRI dataset. As a preliminary analysis, we examined the eigenvalue spectrum of the first PCA eigenconnectivity matrix B_PCA to roughly estimate a reasonable number of modules K (Fig 2). Note that the number of modules in MCF equals the number of eigenvalues retained in the rank-K approximation of B_PCA ≈ UQU^⊤ performed in the generalized version of Algorithm 1 (Section 2.3.5). The figure suggested that the first three (84%) or four modules (89%) already cover a large fraction of variability explained by B_PCA. We thus performed MCF with K = 3 and K = 4, extracting only one component in each case for simplicity. To avoid local optima, we again used 20 different initial conditions in each K, and examined the results that achieved the maximum explained variance.

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Fig 2. Resting-state fcMRI data.

Top 20 squared eigenvalues (black dashed line) of first PCA eigenconnectivity matrix B_PCA. Their cumulative proportions (with respect to total of 177 eigenvalues) are shown in right vertical axis (blue solid line).

https://doi.org/10.1371/journal.pone.0168180.g002

2.5.3 Visualization.

To visualize the spatial weight patterns, we used a simple scheme adopted [18]. We plotted a dot at each spatial coordinate of the ROI in different sizes and colors and displayed the 3D spatial patterns in three different views: from the right lateral, the dorsal, and the posterior. In each view, we projected the 3D dots to the three 2D planes, rather than sectioning the 3D brain, so that the brain is regarded as transparent.

In addition, we illustrated module-level eigenconnectivity G as an undirected graph with self-loops, where the squared magnitude and the sign of each entry are indicated by the edge width and color. The OCF graph was also drawn in a similar manner, while G was actually a constant matrix with zero diagonals. Here, we set the global sign of each G (which is arbitrary) to maximize the sum of the squares of the positive entries. Due to the arbitrariness of the global sign, the signs (and colors) in G do not indicate excitatory or inhibitory relationships between modules. Rather, they imply that the module-level connectivities of different signs are mutually anti-correlated when varying over individuals.

3.2.1 Comparison with OCF.

First we show the results by MCF of K = 2 in comparison to OCF. Fig 5(a) and 5(b), compare the first and second components, each of which were both obtained by MCF and OCF. These components explain the inter-individual variabilities in the connectivity matrices. The order of the two weight vectors in each method is actually arbitrary; we reordered the two OCF’s weight vectors to maximize their absolute inner products with the corresponding MCF’s weight vectors. The signs in the OCF’s weight vectors were also appropriately flipped so that the patterns are comparable between the two methods.

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Fig 5. Resting-state fcMRI data.

Spatial patterns of weight vector pairs, w₁ and w₂, as well as eigenconnectivity G between the two modules, obtained as (a) first and (b) second components of MCF and OCF. Dots display the magnitudes (indicated by color intensity and area and rescaled in each weight vector) and signs (red: positive, blue: negative) of weight values at 177 ROIs. The brain is transparently viewed from right lateral, dorsal, and posterior side. At center of each pair, eigenconnectivity G is represented as a two-node undirected graph. Magnitude of each entry is indicated by width of edge (straight bars: off-diagonal entries, loops: diagonal entries); signs are indicated by two colors (orange: positive, purple: negative).

https://doi.org/10.1371/journal.pone.0168180.g005

In both components, each spatial pattern obtained by MCF apparently consists of a few spatially localized submodules.

In the first component (Fig 5(a)) of MCF, w₁ forms an interesting network of primary sensory areas, mainly of visual and sensorimotor modalities, with additional participation from the auditory areas. This network is characterized by a large variability of intra-module connectivity, reflected in a large diagonal entry in G (see below).

The w₂ network in the first MCF component is characterized by large weights on the subcortical areas (thalamus and basal ganglia), while other areas in the frontal cortex, the posterior cingulate cortex (PCC), and the angular gyri (AG) also participated. We do not have a clear interpretation for this network, which seems to combine elements of the executive control and the default-mode network (DMN) with subcortical structures.

In the second MCF component (Fig 5(b)), w₁ strikingly resembles the DMN, including ACC, AG, and the medial prefrontal cortex as well as parts of the medial temporal lobe. w₂ seems to be a combination of the task-positive network (TPN) [28] and the salience network [29].

The spatial patterns of the positive weights (red) in OCF resemble those of MCF. However, the OCF patterns combine negative weights (blue) and positive weights, which may complicate interpretation. The negative weights of w₁ in the first component again resemble the DMN, while the positive weights resemble the general sensory areas network in MCF. In w₂, the positive parts appear to form the salience network, while the negative parts may be a purely visual network. Thus, in this result, each OCF pattern seems to simultaneously represent two distinct functional (sub)networks in the positive and negative parts of the weight vector. Readers might wonder why the OCF results look different from the original results [18], obtained for a subset (103 subjects) of our full dataset. However, the difference is actually not qualitatively large after appropriately flipping the signs (specifically those of w₁ in Fig 5(a) and w₂ in Fig 5(b) of our OCF results. Note that the signs of the weight vectors are arbitrary in OCF due to the inherent indeterminacy.

The module-level eigenconnectivity, G, illustrated as a two-node undirected graph at the center, further indicates that the variability within some modules was not negligible. The different signs between the intra-and inter-module eigenconnectivity in both the first and second MCF components mean that these component explains anti-correlated changes between the two types of connectivities. As mentioned above, the first MCF component (Fig 5(a)) exhibited almost the same order of intra-module variability (in the module represented by w₁) as that of the variability of inter-module connectivity . This is easy to understand since the network in the w₁ groups together sensory areas of different modalities, and the connectivities among the sensory modalities are probably quite variable. In contrast, as illustrated by the simulation study in Section 2.4.1, we must be careful about the interpretation of the OCF results when the intra-module network variability is not negligible.

3.2.2 Further demonstration with more than two modules.

Our MCF can be generalized with more than two modules (Section 2.3.5). Here, we see that using a higher K may increase the neurophysiological interest and ease interpretation.

We first show the weight and module-level eigenconnectivity patterns for K = 3 (Fig 6(a)). First, w₁ seems to represent the DMN, possibly with some additional temporal contribution. Second, w₂ seems to be a general sensory area network, similar to w₁ in Fig 5(a), with slightly less emphasis on the visual and anterior sensorimotor areas. Finally, w₃ resembles the salience network [29] with a particular contribution from the insula.

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Fig 6. Resting-state fcMRI data.

Spatial patterns of weight vectors w_k as well as module-level eigenconnectivity G, obtained as first components of MCF with number of modules (a) K = 3 and (b) K = 4. See caption of Fig 5 for visualization details.

https://doi.org/10.1371/journal.pone.0168180.g006

Thus, the three networks seem to represent clearly defined networks, two of which are well-known (salience and DMN) and one of which is new but reasonable (sensory areas). But what is even more interesting is that this particular combination of three networks appears very logical in the following sense; the sensory area network is related to outward-directed information processing, and the DMN is related to inward-directed information processing, while the salience network switches between the two, in particular, detecting when new significant sensory information starts arriving. In fact, the connectivities of the salience network to the DMN and the sensory network have opposite signs.

When the number of modules (networks) is increased to K = 4 (Fig 6(b)), the three networks found for K = 3 are mainly intact, but we see a new network in w₄ that seems to be a very clear-cut TPN with contributions around but not including the central sulcus, similar to [28]. It is also interesting that the DMN’s intra-module variability is rather large compared to all the others. This is presumably due to its different parts (e.g., anterior vs. posterior parts) that perform different functions and thus with variable connectivities [5, 30]. Additional results by Stepwise MCF and another stepwise analysis using k-means clustering is given in S2 Appendix; a further comparison with an existing nonnegative tensor factorization method is found in S3 Appendix.

In this analysis, the running time of MCF to extract the 1st component (K = 4) was 1.55 ± 0.01s (mean±SD, 20 runs) on a Linux computer with 18 cores, 2.30GHz CPU and 128GB RAM, while performing only Stepwise MCF took 1.27 ± 0.01s.