## 2 Background  
In ECG processing, a number of researchers have proposed the use of PCA or ICA to detect clinically relevant waveforms that cannot be straightforwardly observed visually. The applications in this field could be divided into four main groups according to their goal, namely, noise reduction, atrial fibrillation (AF) detection, fetal ECG decoupling, and breathing signal isolation. The first group of works is composed by algorithms implementing ICA or PCA to reduce noise on the ECG. In [8], the authors used the auto-correlation function of the ICA output applied over measured limb leads (I and II) of 9 healthy subjects. The component with the lowest value of area under the curve was reported as the noisiest component, and the signals without noise were computed by subtracting this noisy contribution to the original signals. A PCA approximation to the denoising problem was proposed also in [9], where the authors used a database consisting of normal ECGs artificially contaminated with noise. The algorithm presented included two steps. First, twelve-lead ECGs were detached by beats and compiled into matrices, one per each beat, hence making up one beat of every single lead. PCA was then applied to these matrices, those components achieving 95% of the standard deviation were selected, and then the PCA transformation was inverted. In the second step, the previous matrices were recomputed using the denoised beats, and then PCA was applied over these new matrices by selecting those components achieving the 98% of the standard deviation. Finally, PCA anti-transform was applied to the components computed before, hence obtaining the denoised signals. It can be summarized that these algorithms did not use the multivariate decomposition on a physiological basis, but rather under a computational and statistical rationale. The second group of works is composed by algorithms using ICA or PCA in order to extract the AF signal from the ECG. A PCA approximation to solve this problem was proposed in [10], where the authors presented an algorithm that can quantify the presence of AF, using for that purpose a database compounded by six twelve-lead ECG registers with 300 s each. The registers were divided in two sets of 150 s, and each set was divided again in subsets of 10 s. Each subset was PCA-processed to extract the AF frequency by searching in the frequency domain in every principal component. Finally, the mean and the standard deviation of the AF frequency were computed for the complete record and the 150-second subset. An ICA approximation to solve this problem was proposed in [11], where the authors presented an algorithm to breakdown beats into the classification as normal, AF, and atrial flutter. The algorithm was tested over the MIT-BIH AF database. The beats were firstly detected using the Pan-Tompkins algorithm, and then isolated with a 150-sample window centered on the R-wave. A high order bispectrum was computed for each beat in order to enhance the hidden complexities present in nonlinear signals, and ICA was then applied to the bispectrum output to extract features from each beat. Consecutively, these features were processed with a K-Nearest-Neighbours classifier in order to assign beats to the defined classes. In summary, the multivariate exploration of conventional ECG leads, ICA, or PCA, opens the door to the physiological analysis as it transforms the geographical projections of standard ECG, as well as to a new domain more focused on the originating signal itself, and therefore bundled to the physiological reality. The third group of works consists of algorithms which use ICA to derive the fetal ECG from the recordings of the mother ECG. An approximation to solve this problem is proposed in [12], where the signal was firstly filtered to reduce the noise in the ECG, and then processed by using a compressive sensed sparse binary matrix, which transforms the signal in a set of words from a previously created dictionary.