Tracing Evolving Networks Using Tensor Factorizations vs. ICA-Based Approaches

Abstract

Analysis of time-evolving data is crucial to understand the functioning of dynamic systems such as the brain. For instance, analysis of functional magnetic resonance imaging (fMRI) data collected during a task may reveal spatial regions of interest, and how they evolve during the task. However, capturing underlying spatial patterns as well as their change in time is challenging. The traditional approach in fMRI data analysis is to assume that underlying spatial regions of interest are static. In this article, using fractional amplitude of low-frequency fluctuations (fALFF) as an effective way to summarize the variability in fMRI data collected during a task, we arrange time-evolving fMRI data as a subjects by voxels by time windows tensor, and analyze the tensor using a tensor factorization-based approach called a PARAFAC2 model to reveal spatial dynamics. The PARAFAC2 model jointly analyzes data from multiple time windows revealing subject-mode patterns, evolving spatial regions (also referred to as networks) and temporal patterns. We compare the PARAFAC2 model with matrix factorization-based approaches relying on independent components, namely, joint independent component analysis (ICA) and independent vector analysis (IVA), commonly used in neuroimaging data analysis. We assess the performance of the methods in terms of capturing evolving networks through extensive numerical experiments demonstrating their modeling assumptions. In particular, we show that (i) PARAFAC2 provides a compact representation in all modes, i.e., subjects, time, and voxels, revealing temporal patterns as well as evolving spatial networks, (ii) joint ICA is as effective as PARAFAC2 in terms of revealing evolving networks but does not reveal temporal patterns, (iii) IVA's performance depends on sample size, data distribution and covariance structure of underlying networks. When these assumptions are satisfied, IVA is as accurate as the other methods, (iv) when subject-mode patterns differ from one time window to another, IVA is the most accurate. Furthermore, we analyze real fMRI data collected during a sensory motor task, and demonstrate that a component indicating statistically significant group difference between patients with schizophrenia and healthy controls is captured, which includes primary and secondary motor regions, cerebellum, and temporal lobe, revealing a meaningful spatial map and its temporal change.

Keywords: PARAFAC2; evolving networks; independent component analysis (ICA); independent vector analysis (IVA); spatial dynamics; tensor factorizations; time-evolving data.

Figure 1

Illustration of modeling time-evolving data in the form of a subjects by voxels by time tensor using PARAFAC2, IVA and joint ICA. Following the notation in the literature on ICA/IVA, we use S_k, for k = 1, …, K, to denote the factor matrix in the voxels mode for joint ICA and IVA, where ${\text{S}}_k={\text{B}}_k^T$.

Figure 2

True factors used to generate simulated data. (A) Subject-mode factors, where a_r indicates the columns of A, for r = 1, 2, 3, and (B) Temporal patterns, where c_r indicates the columns of C, for r = 1, 2, 3. Time, here, is in the resolution of time windows, but may also correspond to time samples depending on the application.

Figure 3

True evolving components (R = 3), where each component corresponds to a column of B_k, for k = 1, …, 50, used to generate simulated data with (A) evolving networks, (B) overlapping networks, (C) similar networks.

Figure 4

Case 1a. Evolving components (B_k or S_k, for k = 1, …, K) captured by each method after fixing the scaling and permutation ambiguity: (A) PARAFAC2, (B) IVA, (C) joint ICA. All methods recover the underlying evolving components accurately.

Figure 5

Case 1a (K = 20). p-values obtained using the two-sample t-test on the subject-mode patterns (A or A_k) using different methods as the number of voxels (i.e., J) changes, where the number of time slices (i.e., K) is set to 20. Based on the true subject-mode patterns, true p-values are 0, 0.88, and 0.35 for component 1, 2, and 3, respectively. For large sample size, i.e., J = 10, 000, all methods can identify that the first component is the statistically significant one in terms of group difference. As J decreases, in addition to the first component, IVA returns small p-values for other components in some windows corresponding to false-positive cases while PARAFAC2 and joint ICA work well regardless of the sample size.

Figure 6

Covariance matrices of size K by K showing the covariance structure of true components across K = 50 time slices for (A) evolving networks, (B) overlapping networks, (C) similar networks.

Figure 7

Case 2. Evolving components (B_k or S_k, for k = 1, …, K) captured by each method: (A) PARAFAC2, (B) IVA, (C) joint ICA. PARAFAC2 and joint ICA fail to capture the underlying networks while IVA can reveal the evolving components accurately.

Figure 8

Case 2. p-values obtained using the two-sample t-test on the subject-mode patterns (A_k) using IVA. IVA successfully captures that the first component is statistically significant in terms of group difference in every other window. In the other components, in some time windows, there are false positives marked with a red arrow.

Figure 9

Case 3. (A) p-values obtained using the two-sample t-test on the subject-mode patterns (A or A_k) using different methods. Based on the true subject-mode patterns, true p-values are 0, 0.88, and 0.35 for component 1, 2 and 3, respectively. All methods identify the first component as the component differentiating between the subject groups. While PARAFAC2 and joint ICA identify the second and third components as not statistically significant in terms of group difference, IVA wrongly identifies them as statistically significant in some windows. (B) Temporal patterns, i.e., columns of factor matrix C, extracted from the time mode using PARAFAC2. True patterns are shown using dashed lines. PARAFAC2 correctly captures the true temporal patterns.

Figure 10

PARAFAC2 analysis of task fMRI data. (A) Spatial components, i.e., columns of B_k. Here, we plot columns of only B₁ corresponding to the first time window. The corresponding p-values are 7.8 × 10⁻⁶ for component 1, and 7.7 × 10⁻¹ for component 2. The first component includes primary and secondary motor and cerebellum, as well as auditory cortex expected to be engaged by the task. Spatial maps are plotted using the patterns from the voxels mode as z-maps and thresholding at |z| ≥ 1.5 such that red voxels indicate an increase in controls over patients, and blue voxels indicate an increase in patients over controls. (B) Temporal patterns, i.e., columns of matrix C.

Figure 11

IVA analysis of task fMRI data. (A) Spatial components, i.e., rows of S_k. Here, we only plot two of the rows of S₁ corresponding to the first time window. Spatial maps are plotted using the patterns from the voxels mode as z-maps and thresholding at |z| ≥ 1.5 such that red voxels indicate an increase in controls over patients, and blue voxels indicate an increase in patients over controls. (B) p-values for the two components in each time window. While component 5 is statistically significant in all but one time window (i.e., time window 3), component 12 is not in any of the time windows.

Figure 12

Joint ICA analysis of task fMRI data. Spatial components, i.e., rows of S_k. Here, we only plot rows of S₁ corresponding to the first time window. Spatial maps are plotted using the patterns from the voxels mode as z-maps and thresholding at |z| ≥ 1.5 such that red voxels indicate an increase in controls over patients, and blue voxels indicate an increase in patients over controls. The p-values are 1.1 × 10⁻⁴ for component 1, and 1.4 × 10⁻¹ for component 2.

Figure 13

PARAFAC2 analysis of (A) only task windows: Spatial components, i.e., columns of B_k, for k = 1, as well as the temporal patterns, i.e., columns of C. The p-values are 2.1 × 10⁻⁴ for component 1, and 2.7 × 10⁻¹ for component 2. (B) Only rest windows: Spatial components, i.e., columns of B_k, for k = 1, as well as the temporal patterns, i.e., columns of C. The p-values are 6.6 × 10⁻³ for component 1, and 8.5 × 10⁻¹ for component 2. The first component shows statistical significance in terms of group difference in both task and rest windows; therefore, supporting the modeling assumptions of PARAFAC2 and joint ICA. Spatial maps are plotted using the patterns from the voxels mode as z-maps and thresholding at |z| ≥ 1.5 such that red voxels indicate an increase in controls over patients, and blue voxels indicate an increase in patients over controls.