# A Novel Hybrid Approach to Deal with DVL Malfunctions for Underwater Integrated Navigation Systems
## 3 Proposed PLSR-SVR Hybrid Predictor
### PLSR for the Prediction of the DVL Measurements  
PLSR was first proposed by [PERSON] et al. in 1983 [29]. As a multivariate statistical analysis method, PLSR is mainly used for modeling between multi-independent variables and multi-dependent variables. By decomposing and selecting the data information, PLSR identifies useful information and noise to overcome the problems caused by the intense relativity between independent variables. The algorithm of PLSR is introduced below. There are \(p\) independent variables (\(x_{1},\ldots,x_{p}\)) and \(q\) dependent variables (\(y_{1},\ldots,y_{q}\)). By observing \(l\) sample points, a data matrix of independent variable \(X=[x_{1},\ldots,x_{p}]_{l\times p}\) and a data matrix of dependent variable \(Y=[y_{1},\ldots,y_{q}]_{l\times q}\) are obtained. The aim of PLSR is to form the structure between \(X\) and \(Y\), and to predict \(Y\) via \(X\)[30]. The first latent vector \(t_{1}\) and \(u_{1}\) are extracted from \(X\) and \(Y\) respectively, and the principles of the extraction are shown as follows. 1. \(t_{1}\) and \(u_{1}\) should carry as much the diversity information of the corresponding data matrix as possible. 2. The degree of correlation between \(t_{1}\) and \(u_{1}\) should reach its maximum. Once the above principles are satisfied, \(X\) and \(Y\) will be well represented by \(t_{1}\) and \(u_{1}\) respectively. Besides, \(t_{1}\) would have the strongest interpretation for \(u_{1}\). Then \(X\) and \(Y\) can be expressed with \(t_{1}\). If the regression achieves satisfactory precision, the algorithm ends. Otherwise, the second latent vector needs to be extracted from the residual matrix. Repeat the process until an acceptable accuracy is obtained. Suppose \(m\) latent vectors (that is, \(t_{1},\ldots,t_{m}\)) have been extracted, the dependent variables \(y_{k}\) (\(k=1,\,2,\ldots,q\)) can be expressed with \(t_{1},\ldots,t_{m}\). By the transition, \(y_{k}\) (\(k=1,\,2,\ldots,q\)) can be expressed with \(x_{1},\ldots,x_{p}\) in the end. The above algorithm is formalized as follows. Step 1: The data matrices \(X\) and \(Y\) are standardized (mean-centered and scaled to unite variances) as  
\[E_{0}=\left(E_{01},\ldots,E_{0p}\right)_{\times p},\,F_{0}=\left(F_{01},\ldots,F_{0q}\right)_{\mid\times q} \tag{16}\]  
Step 2: Calculate the weight vector \(w_{1}\) which is the eigenvector corresponding to the greatest eigenvalue of the matrix \(E_{0}{}^{T}F_{0}F_{0}{}^{T}E_{0}\). Step 3: Calculate the latent vector \(t_{1}\). \[t_{1}=E_{0}w_{1} \tag{17}\]  
Step 4: Calculate the loading vectors \(p_{1}\) and \(r_{1}\). \[p_{1}=\frac{E_{0}{}^{T}t_{1}}{\left\|t_{1}\right\|^{2}},r_{1}=\frac{F_{0}{}^{ T}t_{1}}{\left\|t_{1}\right\|^{2}} \tag{18}\]  
Step 5: Calculate the residual matrices \(E_{1}\) and \(F_{1}\). \[E_{1}=E_{0}-t_{1}{p_{1}}^{T},F_{1}=F_{0}-t_{1}{r_{1}}^{T} \tag{19}\]  
Step 6: If the stopping criterion is met, end the iteration. Otherwise, replace \(E_{0}\) and \(F_{0}\) with \(E_{1}\) and \(F_{1}\), and go to Step 2. Suppose the iteration stops at the \(l\)th iteration, and \(h\) latent vectors (that is, \(t_{1},\ldots,t_{h}\)) are obtained. \(F_{0}\) can then be expressed with the latent vectors as  
\[F_{0}=t_{1}{r_{1}}^{T}+\cdots+t_{h}{r_{h}}^{T}+F_{h} \tag{20}\]  
where \(F_{h}\) is the residual matrix of \(F_{0}\) when the \(h\) latent vectors are included in the PLSR method. By using Equations (17) and (20), an expression for \(F_{0}\) in terms of \(E_{0}\) can be derived by means of the mathematical induction. \[F_{0}=E_{0}\Bigg{\{}w_{1}{r_{1}}^{T}+\sum_{i=2}^{h}\left(\prod_{j=1}^{i-1} \left(I-w_{j}{p_{j}}^{T}\right)\right)w_{i}{r_{i}}^{T}\Bigg{\}}+F_{h} \tag{21}\]  
Finally, the regression equation of \(Y\) about \(X\) with regard to the PLSR model can be obtained according to Equation (21). When the DVL is working properly, the training samples for PLSR contain the SINS calculating results (specifically, velocities along the navigation frame) as inputs and the DVL measurements as desired outputs.