# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 3 Applications
### Signal Processing  
In [32] the applications of a multi-class SVM and a complex SVM were presented [14] to classify four types of human heartbeat using electrocardiogram signals (ECG) as the input data. The purpose of this application was to aid in the diagnosis of arrhythmia. The complex kernel function presented in [14] was used, and a preprocessing of the ECG beat signal employing a discrete Fourier Transform (DFT) was performed. Accuracies between 86% and 94% were obtained. Additionally, a discussion of the extension of input and output spaces of arbitrary dimensions using Clifford SVM [9] was presented. The direction of arrival (DOA) estimation problem is a signal processing problem that deals with the obtention of the direction from which a propagating wave arrives at a set of sensors (sensor array). In [33] an approach to solve the DOA problem using a reformulation of one of the most used DOA algorithms, the Minimum Variance Distortionless Response (MVDR) [34] combined with an SVM extension that \"can be viewed as a particular case of CSVM [9]\" is presented to obtain the SVM-MVDR algorithm. Then, a theoretical relationship (an equivalency) between the sample-based estimation of the output power spectrum of each filter at a certain frequency (SBE) computed by the MVDR and the SBE and estimated by the MUltiple Signal Classification (MUSIC) [35; 36] is derived by analyzing the high-resolution nonparametric spectrum estimation procedure MVDR. This equivalency is possible when matrices corresponding to the signal space of the MVDR are dropped and the noise eigenvalues are changed by ones. Thus, using the above relationship, an SVM-MUSIC is proposed to solve the DOA estimation problem. As the performance of the standard MUSIC and MVDR, and the SVM algorithms to solve DOA, are reduced when they deal with coherent signals, the authors proposed the application of another solution for the DOA problem, the spatial smoothing (SS) technique [37] to \"provide algorithms with coherent signal detection abilities\". SS divides the sensor array into subarrays, performs individual algorithms in the subarrays, and then calculates the average of the estimated spectra. The experimental results show that the SVM-MUSIC and SVM-MVDR combined with SS obtained the best recognition results compared to their classic versions. The beamforming problem involves the estimation of a signal from a given direction. It is usually defined as the most prominent technique to estimate DOA. In [38], a proposal to solve beamforming by modification of the classic MVDR or Capon beamformer [39; 40] is presented, including a regularization term, the so-called \(\epsilon\)-insensitive loss function that penalizes sidelobe levels and allows a certain error in the desired signal array response direction. The result is a convex optimization problem with linear restrictions, which is equal to the SVR problem. Although initially the SVR-MVDR problem was defined in the complex domain, where inputs, outputs, and weights are complex-valued vectors and matrices, this formulation was later transformed from complex to real domains by splitting the real and imaginary parts of the desired beamformer output and, the transformed steering vectors, obtaining a split-definition for the beamformer weights. The proposed method was shown to provide \"suitable results\" and to outperform Capon and SpheRCB beamformers [40], especially in environments where the signal-to-noise ratio (SNR) is high, the number of available snapshots is limited, and the calculation of the signal covariance matrix is a difficult task for classic approaches. Ground penetrating radar (GPR) is a tool that uses radar pulses to image subsurfaces, and it is a very useful sensor in the fields of defense, civil engineering, environment, agriculture, among others. \"In civil engineering, the information of the vertical structure of the stratified media can be extracted from radar profiles using echo detection and amplitude estimation.