# A Novel Hybrid Approach to Deal with DVL Malfunctions for Underwater Integrated Navigation Systems
## 3 Proposed PLSR-SVR Hybrid Predictor
### SVR for the Prediction of the Residual Components  
Since PLSR is linear, the predictor can only handle the linear relationship between the independent variables and the dependent variables. In order to improve the prediction accuracy, SVR is introduced to predict the residual components of PLSR. SVR is based on extension of the support vector concept proposed by [PERSON] et al. (2017). The aim of SVR is to provide a nonlinear mapping function to map the training data to a high dimension feature space ([PERSON] et al., 2017). The algorithm of SVR is introduced below. There is a sample set \(S=\{(x_{i},y_{i})\}\), where \(x_{i}\) and \(y_{i}\) are the input vector and the target output respectively. In the high dimension feature space, there theoretically exists a linear function \(f\) to formulate the nonlinear relationship between \(x_{i}\) and \(y_{i}\)([PERSON] et al., 2017). The function is called the SVR function, which is defined as  
\[f(x)=v^{T}\phi(x)+b \tag{24}\]  
where \(v\) is the weight vector. \(\phi(x)\) is the mapping function to map \(x\) to a high dimension feature space. \(b\) is the bias term. The training error of SVR is determined by the \(\varepsilon\)-insensitivity loss function which is defined as follows. \[\left|y-f(x)\right|_{\varepsilon}=\left\{ \begin{array}{cc}0&\left|y-f(x) \right|\leq\varepsilon\\ \left|y-f(x)\right|-\varepsilon&\text{otherwise}\end{array} \right. \tag{25}\]  
where \(\varepsilon\) is the insensitivity loss zone. The \(\varepsilon\)-insensitivity loss function can improve the robustness of the regression. Different from the traditional regression models that estimate the coefficients through minimizing the loss square, SVR aims to minimize the empirical and structural risk. Then the regression problem is equivalent to the problem of minimizing the cost function which is described as  
\[\min\frac{1}{2}v^{T}v+C\sum_{i=1}^{m}\left(\xi_{i}+\xi_{i}^{*}\right) \tag{26}\]  
\[s.t.\left\{ \begin{array}{cc}y_{i}-v^{T}\phi(x_{i})-b\leq\varepsilon+\xi_{i} \\ v^{T}\phi(x_{i})+b-y_{i}\leq\varepsilon+\xi_{i}^{*}\\ \xi_{i}\geq 0,\ \xi_{i}^{*}\geq 0\end{array} \right. \tag{27}\]  
where \(m\) is the number of the training sample. \(\frac{1}{2}v^{T}v\) is the structural risk to prevent overtraining error. \(\xi_{i}\) and \(\xi_{i}^{*}\) are the positive slack variables which are used to measure the deviation of insensitive boundaries. \((\xi_{i}+\xi_{i}^{*})\) is the empirical risk. \(C\) is a constant to balance between the empirical and structural risk. By using the method of Lagrange multiplier and the Karush-Kuhn-Tucker condition, the dual problem of the Formula (27) can be described as  
\[\max L(\alpha,\alpha*)=-\frac{1}{2}\sum_{i,j=1}^{m}\left(\alpha_{i}-\alpha_{i} ^{*}\right)(\alpha_{j}-\alpha_{j}^{*})K(x_{i},x_{j})-\varepsilon\sum_{i=1}^{m }\left(\alpha_{i}+\alpha_{i}^{*}\right)+\sum_{i=1}^{m}y_{i}(\alpha_{i}-\alpha_ {i}^{  
Finally, the SVR function can be obtained as  
\[f(x)=\sum_{i=1}^{m}{(\alpha_{i}-\alpha_{i}^{*})K(x,x_{i})}+b \tag{31}\]  
where \(K(x,x_{i})\) is the kernel function with the value of the inner product \(<\)\(\phi(x)\), \(\phi(x_{i})\)\(>\). Kernel functions, including polynomial kernel function, radial basic function and Sigmoid kernel function are often used. In this paper, the radial basic function is chosen to be the kernel function, which is expressed as  
\[K(x,x_{i})=\exp(-|x-x_{i}|^{2}/\sigma^{2}) \tag{32}\]  
where \(\sigma\) is the width of the radial basis function. When the DVL is working properly, the SINS calculating results are taken as the input for the PLSR predictor. The predictor outputs the prediction of the DVL measurements \(\hat{V}_{DVL\_P}^{n}\). The residual components of the PLSR prediction can be calculated by  
\[\delta V_{DVL}^{n}=V_{DVL}^{n}-\hat{V}_{DVL\_P}^{n} \tag{33}\]  
where \(V_{DVL}^{n}\) is the velocity measurements of the DVL.