# Decomposition Methods for Machine Learning with Small, Incomplete or Noisy Datasets
## 3 Results
### Brain Signal Classification
#### 3.1.1 BCI with Missing/Corrupted Measurements  
In BCI applications, noisy or missing data can arise. This can occur due to the lack of connection between wireless EEG headset and the computer, or because artifacts appear due to muscle movements, eye movements or electromagnetic interference, among others. Since EEG measurements can be organized as multidimensional datasets, in [57] a tensor completion approach was proposed, which consists in fitting a tensor decomposition model to the available clean measurements and then infer the noisy or missing parts based on those models (see Figure 2a). The advantage of using tensor methods, compared to classical interpolation algorithms, lies in the ability of these models to handle multidimensional information, in other words, they can capture the intricate relationship among entries in a multidimensional signal. For example, to infer a missing entry in an EEG data tensor, these methods can efficiently exploit the available information in other channels, time samples and trials.  
Several tensor decomposition models and tensor completion algorithms were compared on a freely available dataset ([[http://mon.uvic.cat/data-signal-processing/software/](http://mon.uvic.cat/data-signal-processing/software/)]([http://mon.uvic.cat/data-signal-processing/software/](http://mon.uvic.cat/data-signal-processing/software/))) in [57], which are  
Figure 2: Training a BCI classifier (LDA) with noisy/missing EEG measurements. (**a**) Preprocessing steps: first, the positions in which the data is missed/corrupted are identified; then, a mask is created to ignore values in those positions; and finally, the tensor model reconstructs the missing data. (**b**) Results with randomly missing entries. (**c**) Results with random missing channels. (Figure adapted from [57]).  
based on the CP model of the whole tensor, such as the CP Weighted Optimization (CP-WOPT) [58], the High accuracy Low Rank Tensor Completion (HaLRTC) [59] and the Bayesian CP factorization (BCPF) for tensor completion [60]; and one method that uses the Sparse Tucker decomposition of every \(6\times 6\times 6\) tensor patch (subtensors), the 3D Patch-based Tensor Completion (3 DPB-TC) [50]. In the latter case, a Kronecker dictionary was first learned from a clean EEG training dataset. For the experiments with incomplete measurements, two different patterns of missing data were considered: random missing entries, and random missing channels, as shown in Figure 2b,c, respectively. The latter case, represent a realistic situation in which some electrodes are simultaneously disconnected during a complete trial.  
The performance of the tensor completion algorithms was compared, together with a simple interpolation strategy as shown in Figure 2b,c. The classification accuracy of imagined movement in a BCI experiment, using a Linear Discriminant Analysis (LDA) classifier, was evaluated in the perfect case (no missing data) and with each one of the recovered missing data through the tensor completion algorithms. Experimental results demonstrated that all tensor completion algorithms were able to recover missing samples increasing the classification performance compared to a simple interpolation approach, because tensor methods are able to exploit the multidimensional correlation of data. As expected, the random missing samples was easier to reconstruct using the neighbour points, while the random missing channels was more difficult because the amount information was missed in the same neighbourhood (the same channel, in that case). Therefore, as experimentally demonstrated, tensor completion algorithms could be used in real BCI data to avoid discarding noisy frames or frames with missing data, and instead recover the missing data using that technique.