# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 3 Applications
### Time-Series Prediction and Function Approximation  
In [23] (Section 2.2.3), using complex-valued SVR called \(\epsilon\)-complex SVR, a function approximation experiment was conducted in which the Laplacian and Gaussian noises were added to a complex function. In the presence of the Laplacian noise, the C-SVR of [PERSON] et al. [7,8] performed better than the \(\epsilon\)-SVR, the LS-SVR, and the M-SVR. Meanwhile, a function added with the Gaussian noise was better-approximated by the \(\epsilon\)-SVR. In addition, in [23], a chaotic time-series prediction task was solved. With the Lorenz attractor [49] data, the authors derived an eight-step predictor with a \(\epsilon\)-quaternion-valued SVR, and the rotationally invariant loss function defined by their approach allows the production of better models compared to those obtained using the C-SVR. It is important to note that the solution to the set of equations that models the Lorenz attractor has several applications in physical systems that show chaotic behavior; thus predicting and identifying chaotic systems using complex- and hypercomplex-valued SVRs is a promising application that could lead to obtaining better accuracies than approaches that use real-valued regressors or predictors. Two experiments in time-series prediction using a recurrent CSVM was presented in [9]. The authors designed a recurrent prediction system that consisted of a long short-term memory (LSTM) [20] neural network connected to a CSVM. The LSTM module solves the problem of identifying temporal dependencies between input and output data; meanwhile, the CSVM provides precision to the prediction system by optimizing the learning of the data mapping. For this, the water levels in the Venice Lagoon during the periods from 1980 to 1989 and 1990 to 1995 were used as the input data. Four-hundred records were used for training and 600 to be predicted by the LSTM-CSVM, the minimum training error was 0.0019. In the second forecasting experiment, the Mackey-Glass time series data was used and the LSTM-CSVM achieved the lower error compared to error of Echo-state networks [50], Evolino [51,52] and LSTM. In addition, in [9], a case of multiple function approximation was solved using a CSVM. Two different functions involving \(sin()\) and \(cos()\) were interpolated at the same time, with a two output CSVM in which 50 points of each function were used to train the CSVM and experiments interpolating 100 and 400 points were performed, obtaining highly accurate values. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Ref./Year** & **Application** & **SVM Extension Used** \\ \hline
[9]/2010 & Object recognition problems using synthetic and real-object data & CSVM of [9] \\ \hline
[48]/2013 & Pattern recognition using iris data and breast cancer data & Extension of a CSVM [9] \\ \hline
[47]/2014 & Object recognition and emotion of clouds in remote-sensing images & Modification of the CSVM of [9] involving a multi-output support vector regression combined with a support vector value contourlet transform \\ \hline
[14]/2015 & Multiclass classification experiments using the MNIST database of handwritten digits. & Complex-valued SVM [14] \\ \hline
[45]/2016 & Survey of advances in multi-channel classification for THz pulse imaging and dynamic contrast-enhanced MRI & Clifford SVM [8,9] \\ \hline
[46]/2016 & Classification of substances based on their THz spectra & GA-SVM \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Applications of complex and hypercomplex SVMs for solving pattern recognition problems. In [14], a complex-valued function such as \(sinc()\) was approximated using a real CSVR and a DRC-SVR (see Section 2.2.5). The results obtained shows that the DRC-SVR \"fails to capture the complex structure of the function\", and, on the other hand, the CSVR estimates the function with a mean MSE that is equivalent to \(-15.75\) dB, vs. \(-10.42\) dB of the DRC-SVR.