# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 3 Applications
### Signal Processing
Echo detection provides the time delay estimation (TDE) associated with each interface (which provides important information about the probed media structure), whereas amplitude estimation is used to retrieve the wave speed within each layer [41].\" In [41] a combined approach of SVR and forward-backward linear prediction (FBLP) [42] is presented to solving it. The objective function of SVR as formulated in the complex domain using Wirtinger's calculus as it is done in [14]. The experimental results using overlapping and nonoverlapping signals and limited snapshots show that FBLP-SVR outperforms standard FBLP and forward linear prediction (FLP) [43]. Furthermore, FBLP-SVR requires only one snapshot to function. Signals of the geo-positioning system (GPS) and multichannel human heart rate during a very long cycling episode are analyzed in [44] to determine the specific biomedical features, obtain a relation between the heart rate and slopes for downhill and uphill cycling, and the mean heart rate evolution on flat segments. The signals are pre-processed using low-pass finite impulse response (FIR) noise-filtering. In addition, uniform resampling, GPS, and Google maps region information data are merged, and segments of signal patterns are selected to fulfill some criteria (altitude gradient within given limits for uphill, downhill, or flat cycling). Then, the cross-covariance function and the correlation coefficient between data segments are computed to select the data segments that have 95% confidence related to the regression lines. Then a three-feature vector is made for each cycling segment and a two-class classification is performed using NN, complex SVM of [PERSON] et al. [14], and the k-nearest neighbor method, to distinguish between the mean altitude less than 1500 m and greater than 1500 m. The experimental results show that a higher classification accuracy was achieved by NN with sigmoidal transfer functions in the first layer and the probabilistic softmax transfer function in the second layer. The output layer provides a probability for each class based on [PERSON] theorem. This is an example of the application of a complex SVM with real-valued data. When the data is not naturally embedded in complex or hypercomplex spaces, the advantages are usually not as great compared to real-valued approaches, such as the NN of this case. In Table 2, the applications mentioned in this subsection to solve signal processing problems are presented in a compact manner. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Ref./Year** & **Application** & **SVM Extension Used** \\ \hline
[38]/2004 & Beamforming problem & Modification of MVDR and a complex-valued SVM. \\ \hline
[33]/2012 & DOA estimation problem & CSVM-MVDR [9], SVM-MUSIC, SS-CSVM-MVDR and SS-SVM-MUSIC \\ \hline
[32]/2016 & Classification of four types of human heartbeat using ECG signals as the input data & Multi-class complex-valued SVM [14] \\ \hline
[44]/2017 & Determination of biomedical features during cycling episodes from GPS and multichannel human heart rate signals & Complex-valued SVM [14] \\ \hline
[41]/2018 & Echo detection and amplitude estimation of GPR signals & Combined approach of a complex-valued SVR and FBLP \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Applications of complex and hypercomplex SVMs for solving signal processing problems.