# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 2 Complex and Hypercomplex Support Vector Machines Extensions
### Division Algebras
#### 2.2.4 A Note on Octonionic Support Vector Regression  
Mentioned in the above section, [PERSON] et al., in their 2010 paper [23], did not include the design of an octonion-valued SVR. Instead, they include this design in a publication in 2012, [13]. Thus, in this extension to octonionic-valued SVR, to derive the primal and dual optimization problems the authors use the same methodology as the one used in [23]. The dual training problem obtained is similar to that presented for quaternionic-valued SVR; however, in the case of octonions, their nonassociativity feature had to be considered to this derivation. Therefore, the dual-training problem considers an antiassociator operator affecting the Kernel matrix term of the dual problem, as well as a substracting quadratic term involving the Lagrange multipliers and an \"associated kernel function\" that computes a pondered \(\epsilon\)-summation of the kernel results and returns a real number. The same happens with the training machine form between the quaternion and octonion-valued SVR, for this last one, a term that it is included as a multiplier of the value of the associated kernel function is involved. Therefore, although their proposals for derivations of quaternionic and octonionic-valued SVRs are similar, the important difference in antiassociativity of octonions as regards the quaternions has to be factored in to compute the dual training problem, as well as the forms of training machine. This differs from the cases of the CSVM Section 2.1 and from the proposal presented in the next subsection, in which the generic and invariant methodologies of derivation of the dual optimization problem are shown independent of the algebra with which one works.  
The paper [13] also shows three special cases of octonionic SVR in which the nonassociative octonion-valued loss-function-related terms can be neglected to obtain a \"full-analogy\" (not identity) between their proposals of real, complex, quaternion, and octonion SVR. These special cases were called pseudoreal, pseudocomplex, and pseudoquaternion regressions. The authors analyzed the features of a nonlinear function that maps data from the input space to the feature space (feature map or kernel function), to describe the cases when the associated kernel function terms are equal to zero, and the full analogy between their octonion-valued and quaternion-valued SVR proposal is achieved. Nevertheless, in the general octonion-valued SVR case, the curse of dimensionality could lead to computationally intractable problems depending on the number of input data, and therefore, this could render it impractical in real applications. However, this was not the case in the experimental results section.  
Experimental results demonstrate the performance of the octonionic SVR when applied to the biomechanical study of human locomotion, i.e., gait analysis. In [28] a high correlation between inertial measurement systems (IMUs: accelerometers and gyroscopes) during human locomotion is suggested. However, due to the fact that the IMUs suffer from intrinsic drift error, there is a need to correct the sensor measurements or systems calibration. One way to achieve this is to use an intelligent regression to estimate the correct measurements of key events of the endpoint foot trajectory during human treadmill walking. This was performed by octonionic SVR, C-SVR, LS-SVR, and M-SVR. The targets of regression were expressed as pure octonions and, as the data sets contain a significant number of outliers, the octonionic SVR and the C-SVR obtained the best overall error, and the first one shows more consistency in obtaining better results.