On the robustness of EEG tensor completion methods
DUAN Feng1∗, JIA Hao1, ZHANG ZhiWen1, FENG Fan1, TAN Ying1, DAI YangYang1, CICHOCKI Andrzej2,3,4,5, YANG
ZhengLu6, CAIAFA Cesar F.1,7∗, SUN Zhe1,8∗ &
SOL´E-CASALS Jordi1,9,10∗
1College of Artificial Intelligence, Nankai University, Tianjin, China
2Skolkowo Institute of Science and Technology, Moscow, Russia
3College of Computer Science, Hangzhou Dianzi University, Hangzhou, China
4Department of Informatics, Nicolaus Copernicus University, Poland
5Systems Research Institute of Polish of Academy of Science, Warsaw, Poland
6College of Computer Science, Nankai University, Jinnan, Tianjin, China
7Instituto Argentino de Radioastronom´ıa - CCT La Plata, CONICET/CIC-PBA/UNLP, Argentina
8Computational Engineering Applications Unit, Head Office for Information Systems and Cybersecurity, RIKEN, Wako-Shi, Japan
9Department of Psychiatry, University of Cambridge, Cambridge, UK
10Data and Signal Processing Research Group, University of Vic - Central University of Catalonia, 08500 Vic, Catalonia, Spain
*Corresponding authors
E-mail: duanf@nankai.edu.cn, ccaiafa@fi.uba.ar, zhe.sun.vk@riken.jp, jordi.sole@uvic.cat
1. Average spectrum error
The LNRMSE measures the error between the completed tensors and the original tensors in the time domain. In the analysis of EEG signals, the frequency domain also plays an important role. To compare the differences between the completion effects in the frequency domain, we adopt the spectrogram as a measurement, which is a photograph or diagram of a spectrum. The spectrogram gives the short-time Fourier transform of the input signal, which is a two-dimensional time-frequency matrix filled with complex values. The average spectrum error measures the differences in the spectrograms between the original and completed tensors. Suppose there exists a tensor with NmCT channels containing missing entries. For each of the NmCT time series, the spectrograms of the original time series and the completed time series are calculated and denoted as Pc and ˆPc, c = 1,2, . . . , NmCT individually. There are three steps when calculating the spectrogram:
1. Divide the signal into sections of length 128, windowed with a Hamming window;
2. Specify 120 samples of overlap between adjoining sections;
On the robustness of EEG tensor completion methods 2 3. Evaluate the spectrum at 1282 + 1=65 frequencies and 768128−−120120=81 time bins.
Because each time series contains original entries, the values of the spectrum are the same. These entries with same values will not be considered in the calculation of errors. If the number of non-zero values in (Pc−Pˆc) is denoted asNnonzeros, the average spectrum error can be obtained using
ASE =
∑NmCT
c=1 abs(Pc−Pˆc)
Nnonzeros×NmCT . (1)
When calculating the differences between two permutations in Simulation II, we also adopt the logarithm average spectrum error, similarly to LNRMSE:
δLASE = −log10(abs(ASE⟨X,Xˆchannel⟩−ASE⟨X,X ⟩ˆ )), (2) or
δLASE = −log10(abs(ASE⟨X,Xˆtrial⟩−ASE⟨X,X ⟩ˆ )). (3) The ASE analysis results obtained on Simulation I and Simulation II are shown in Figure 1, Figure 2 and Figure 3. The main difference between the ASE and the LNRMSE is that the ASE evaluates the differences in the frequency domain whereas the LNRMSE does so in the time domain. Note that a higher LNRMSE means better performance while a smaller ASE indicates the same.
Figure 1. Simulation I: ASE comparison with the change ofNmCT.
On the robustness of EEG tensor completion methods 3
Figure 2. Simulation I: ASE comparison with the change ofNmTL.
Figure 3. Simulation II: ASE comparison on the influences of permutations: shuffling of channel ordering (left) and trial order (right).
