How does PCA affect the capability of ICA to extract interpretable brain and non-brain components?

Blind source separation (BSS) methods such as PCA and Independent Component Analysis (ICA), extract an [m n] “unmixing matrix” W where n is the number of channels and m the number of independent components (ICs) retained so that

where X is the original [n,t] dataset and S has dimension [m,t]. The i^th row of represents the time course of the i^th IC (the IC’s ‘activation’). The “mixing matrix” A (the pseudoinverse of W, A = W⁺) represents, column-wise, the weights with which the independent component (ICA) projects to the original channels (the IC ‘scalp maps’). For sake of simplicity, the terms “IC” will be used below for components of PCA->ICA or ICA-Only origin, PCs for components of PCA-Only origin. Note that the notation for PCA transformation differs from the ICA one, as in PCA-related papers the data X has dimensions [t,n], S_PCA [t,m] and W_PCA [n,m] and therefore S_PCA = XW_PCA. In this notation, the data channels are represented row-wise to adhere to ICA-related notation and to enhance the readability of the manuscript.

If the electrode locations are available, the columns of A can be represented in interpolated topographical plots of the scalp surface (“scalp maps”) that are color-coded according to the relative weights and polarities of the component projections to each of the scalp electrodes. While both decompositions have the same linear decomposition form, PCA extracts components (PCs) with uncorrelated time courses and scalp maps, while ICA extracts maximally temporally-independent components (ICs) with unconstrained scalp maps. As linear decompositions, PCA and ICA can be used separately, or PCA can be used as a preprocessing step to ICA to reduce the dimension of the input space and speed ICA convergence.

Since the scalp maps of most effective brain source ICs strongly resemble the projection of a single equivalent current dipole (Delorme et al., 2012), each component IC_n may be associated with a “dipolarity” value, defined as the percent of its scalp map variance successfully explained by a best-fitting single equivalent dipole model, here computed using a best-fitting spherical four/shell head model (shell conductances: 0.33, 0.0042, 1, 0.33; μS, radii 71, 72, 79, 85) using the DIPFIT functions (version 1.02) within the EEGLAB environment (Delorme and Makeig, 2004; Oostenveld and Oostendorp, 2002):

resvar (IC_n) being the fraction of residual variance explained by the equivalent dipole model,

For ‘quasi-dipolar’ components with dip(IC_n) > 85% and especially for ‘near-dipolar’ components with dip (IC_n) > ~95%, the position and orientation of their equivalent dipole is likely to mark the estimated location of the component source (with an accuracy depending on the quality of the decomposition and the accuracy of the forward-problem head model used to fit the dipole model). As shown in Figure 3 of the component source (with an accuracy depending on the quality of the decomposition and the accuracy of the forward-problem head model used to fit the dipole model). As shown in Figure 3 of (Artoni et al., 2014), ICs with dip(IC_n) > 85% have the lower likelihood of also having a low quality index (meaning they have stability to resampling). In other words, highly dipolar ICs are more likely to be stable than low dipolar ICs. As in (Delorme et al., 2012) and (Artoni et al., 2014), here we define “decomposition dipolarity” as the number of ICs with a dipolarity value higher than a given threshold (e.g., 85%, 95%).

Panels A and B: box plots of median numbers of ICs (#ICs) with dipolarity values (A) above 85% (quasi-dipolar) and (B) 95% (near-dipolar). Significance of differences between conditions was determined using Kruskal-Wallis plus Tuckey post hoc tests. Panel C: Estimated probabilities of significant condition differences in the number of quasi-dipolar components (RV > 85%) for the following comparisons: (i) PCA-Only versus PCA₈₅ICA; (ii) PCA₈₅ICA versus PCA₉₅ICA; (iii) PCA₉₅ICA versus PCA₉₉ICA; (iv) PCA₉₉ICA versus ICA-Only. Each panel shows p-values for existence of significant differences between the number of quasi-dipolar components in the contrasted condition pair for each dipolarity threshold (x axis, RV > 80% to RV>99%). Dashed red lines show the dipolarity condition-difference significance threshold (red dashed line at p=0.05). Panel D: Numbers of dipolar ICs (y axis) available after PCA dimensionality reduction for two dipolarity thresholds (dipolarity > 85%, >95%) in decomposition conditions PCA₈₅ICA (black dots), PCA₉₅ICA (green dots), PCA₉₉ICA (blue dots), and ICA-only (red dots). A dashed blue line connects the dots for each subject. A red dashed line plots the #ICs (the upper bound to the #dipolar ICs).

To test how preliminary principal PCA subspace selection affects the capability of ICA to extract meaningful artifact and brain components from EEG data, we applied ICA decomposition to each subject’s dataset (i) after applying PCA and retaining 85%, 95%, or 99% of the data variance (PCA₈₅ICA, PCA₉₅ICA, PCA₉₉ICA); (ii) by performing ICA decomposition without preliminary PCA (ICA-Only); or (iii) by applying PCA directly with no subsequent ICA (PCA-Only). In each case, we sorted quasi-dipolar ICs (defined here as dip(IC_n) > 85%) into non-brain (“artifact”) and “brain” subsets, depending on the location of the model equivalent dipole. The artifact subspace was mainly comprised of recurring, spatial stereotyped (i.e., originating from a spatially fixed source) neck muscle activities or ocular movements. Example results for one subject are shown in Figure 2.

For a representative subject, scalp maps of quasi-dipolar components (dipolarity above 85%) extracted by applying ICA (ICA-Only) or PCA (PCA-Only) directly to the data, or by performing ICA after reducing the original data rank by PCA so as to retain at least 85% (PCA₈₅ICA, 4 ± 0.5 Median ± MAD PCs), 95% (PCA₉₅ICA, 8 ± 2.5 PCs) and 99% (PCA₉₉ICA, 21 ± 6 PCs) of data variance respectively. Components are sorted into identifiable non-brain Artifact and Brain ICs, separated by the vertical red dashed line. A dashed blue box highlights eye activity-related artifact ICs (vertical EOG and horizontal EOG ICs, respectively) in the PCA₉₅ICA, PCA₉₉ICA, and ICA-Only conditions.