## 2 Materials and Methods
### C-ICT Framework
#### 2.3.1 C-ICT Step 1: ICA  
In the first stage, ICA is separately performed on each feature dataset to obtain maximally independent estimated source estimates, \(\mathbf{S}^{[k]}\), and corresponding subject covariation matrices \(\mathbf{A}^{[k]},k=1,2,\ldots,K\), as shown in Figure 1a. The original ICA model assumes that the number of observations (subjects) is the same as the number of underlying sources. In practice, usually the number of observations is greater than the number of latent sources. Therefore, directly performing ICA on the original feature matrix might result in overfitting due to the effects of additive noise. So, it is desirable to reduce the dimensionality of the data to a lower-dimensional signal subspace and then perform ICA on the extracted signal subspace. Principal component analysis (PCA), a popular dimensionality reduction technique, is used to represent most of the variability across subjects. Let \(N_{k}\) be the order of the signal subspace--i.e., the number of estimated components--for the \(k\)th dataset. We first perform PCA on \(\mathbf{X}^{[k]}\) to obtain dimension reduced matrices, \(\mathbf{Y}^{[k]}\in\mathbb{R}^{N_{k}\times V_{k}}\), using the following:  
\[\mathbf{Y}^{[k]}=\mathbf{V}^{T[k]}\mathbf{X}^{[k]}, \tag{7}\]  
where \(\mathbf{V}^{T[k]}\in\mathbb{R}^{N_{k}\times M}\) is the dimension reduction matrix. The ICA model applied to \(\mathbf{Y}^{[k]}\) can be written as  
\[\mathbf{Y}^{[k]}=\mathbf{A}_{*}{}^{[k]}\mathbf{S}^{[k]}, \tag{8}\]  
where \(\mathbf{A}_{*}{}^{[k]}\in\mathbb{R}^{N_{k}\times N_{k}}\) denotes the mixing matrix in the ICA model that is in the dimension reduced space. To obtain the back-reconstructed mixing matrix \(\mathbf{A}^{[k]}\in\mathbb{R}^{M\times N_{k}}\) for the original mixture \(\mathbf{X}^{[k]}\), we combine Equations (4), (7) and (8) to obtain the following equation:  
\[\mathbf{A}_{*}{}^{[k]}=\Big{(}\mathbf{V}^{[k]}\Big{)}^{T}\mathbf{A}^{[k]}, \tag{9}\]  
Figure 1: Flowchart of C-ICT. (**a**) Perform ICA on each dataset separately to obtain subject covariation matrices and ICs. (**b**) Eliminate subject covariations corresponding to artifact components. (**c**) Apply IVA-G on the reduced subject covariation matrices to obtain SCVs and the second-level mixing matrices. SCVs that show significant pair-wise correlation are chosen for further analysis. (**d**) Subject covariations with the highest contribution to the correlated SCVs chosen above are identified and their corresponding ICs are identified as the associated components across \(K\) modalities. The first SCV is highlighted for clarity. The same color of the bars in the different matrices denotes the same index in the corresponding rows and columns.  
which is equivalent to  
\[\mathbf{A}^{[k]}=\left(\mathbf{V}^{[k]}\right)^{+T}\mathbf{A}_{*}{}^{[k]}. \tag{10}\]  
where \((\cdot)^{+T}\) denotes the transpose of the Moore-Penrose pseudo-inverse of a matrix. Note that this C-ICT framework allows for different orders for each dataset--an important feature for multimodal data fusion--as shown in Figure 2.  
Applying ICA to each dataset, \(k\), results in a unique source matrix \(\mathbf{S}^{[k]}\in\mathbb{R}^{N_{k}\times V_{k}}\) and estimated mixing matrix \(\mathbf{A}^{[k]}\in\mathbb{R}^{M\times N_{k}}\) for each modality. Note again that the column of \(\mathbf{A}^{[k]}\), denoted by \(\mathbf{a}_{j}^{[k]}\), \(j=1,2,\ldots,N_{k}\), represents the weights of the corresponding source \(\mathbf{S}_{j}^{[k]}\) for each subject and enables the determination of possible connections between different modalities.