ICA's bug: How ghost ICs emerge from effective rank deficiency caused by EEG electrode interpolation and incorrect re-referencing

Independent component analysis (ICA) has been widely used for electroencephalography (EEG) analyses. However, ICA performance relies on several crucial assumptions about the data. Here, we focus on the granularity of data rank, i.e., the number of linearly independent EEG channels. When the data are rank-full (i.e., all channels are independent), ICA produces as many independent components (ICs) as the number of input channels (rank-full decomposition). However, when the input data are rank-deficient, as is the case with bridged or interpolated electrodes, ICA produces the same number of ICs as the data rank (forced rank deficiency decomposition), introducing undesired ghost ICs and indicating a bug in ICA. We demonstrated that the ghost ICs have white noise properties, in both time and frequency domains, while maintaining surprisingly typical scalp topographies, and can therefore be easily missed by EEG researchers and affect findings in unknown ways. This problem occurs when the minimum eigenvalue & lambda;min of the input data is smaller than a certain threshold, leading to matrix inversion failure as if the rank-deficient inversion was forced, even if the data rank is cleanly deficient by one. We defined this problem as the effective rank deficiency. Using sound file mixing simulations, we first demonstrated the effective rank deficiency problem and determined that the critical threshold for & lambda;min is 10-7 in the given situation. Second, we used empirical EEG data to show how two preprocessing stages, re-referencing to average without including the initial reference and non-linear electrode interpolation, caused this forced rank deficiency problem. Finally, we showed that the effective rank deficiency problem can be solved by using the identified threshold (& lambda;min = 10-7) and the correct re-referencing procedure described herein. The former ensures the achievement of effective rank-full decomposition by properly reducing the input data rank, and the latter allows avoidance of a widely practiced incorrect re-referencing approach. Based on the current literature, we discuss the ambiguous status of the initial reference electrode when re-referencing. We have made our data and code available to facilitate the implementation of our recommendations by the EEG community.