## 3 Improved Grey Wolf Particle Swarm Hybrid Optimization MSVR Localization Model Construction  
### Multi-Output Support Vector Regression  
Traditional SVR methods for fingerprint localization require modeling and prediction for each dimension separately. This approach is not only inefficient but also overlooks potential connections between outputs. To surmount these limitations, ref. [42] expanded the one-dimensional SVR to handle multi-dimensional outputs, seeking to address more complex challenges found in real-world engineering scenarios. In this study, for training the MSVR algorithm, we use both x and y coordinates of the reference points as well as the gathered signal features. The goal is to construct a regression model that surpasses the traditional SVR in both localization accuracy and localization efficiency. Let the training sample be denoted as \(S=\{(x_{i},y_{i})|i=1,2,\cdots,K\}\), where \(x_{i}\in\mathbb{R}^{M},y_{i}\in\mathbb{R}^{N}\), \(M\), \(N\), and \(K\) represent the dimensions of the input signal features, the dimensions of the output reference point coordinates, and the number of samples, respectively. The relationship between the reference point coordinates and the signal features can be described as:  
\[\mathbf{y}=\mathbf{P}^{T}\phi(\mathbf{x})+\mathbf{B}=\left[ \begin{array}{c}\mathbf{w}_{1}^{T} \phi(x_{1})+b_{1}\\ w_{2}^{T}\phi(x_{2})+b_{2}\\ \cdots\\ w_{N}^{T}\phi(x_{N})+b_{N}\end{array} \right] \tag{9}\]  
In the equation, \(\phi(\cdot)\) represents the nonlinear mapping in the high-dimensional space, while \(\mathbf{P}=[w_{1},\cdots,w_{j},w_{N}]\) and \(\mathbf{B}=[b_{1},\cdots,b_{j},b_{N}]\) are the weight vector matrix and bias, respectively. The MSVR algorithm substitutes the \(L_{1}\) norm in the traditional single-output SVR's loss function with an \(L_{2}\) norm, which is given by:  
\[L(u)=\left\{ \begin{array}{cc}0&u<\varepsilon\\ (u-\varepsilon)^{2}&u>\varepsilon\end{array} \right. \tag{10}\]where \(L(u)\) represents the loss function defined on the hypersphere, with \(\varepsilon\) being the insensitivity loss domain on the hypersphere. \(u_{i}=\|e_{i}\|=\sqrt{e_{i}{}^{T}_{i}{}_{i}}\), \(e_{i}=y_{i}-P\phi(x_{i})-B\). In the case of multi-output regression, Equation (4) is equivalent to the following constraints:  
\[\text{min}L(\mathbf{P},\mathbf{B})=\frac{1}{2}\underset{i=1}{\overset{N}{\sum}}\left\| \mathbf{w}_{j}\right\|^{2}+C\underset{i=1}{\overset{K}{\sum}}L(u_{i}) \tag{11}\]  
where C is the regularization constant, and \(L\) represents the \(\varepsilon\)-insensitive loss function. By introducing the Lagrange multipliers \(\lambda_{i}\), we obtain the Lagrangian function as:  
\[L(\mathbf{P},\mathbf{B})=\frac{1}{2}\underset{i=1}{\overset{N}{\sum}}\left\|\mathbf{w}_{ j}\right\|^{2}+C\underset{i=1}{\overset{K}{\sum}}L(u_{i})-\underset{i=1}{ \overset{K}{\sum}}\lambda_{i}(u_{i}{}^{2}-\left\|y_{i}-P\phi(x_{i})-\mathbf{B} \right\|^{2}) \tag{12}\]  
As the function resides at an extremum, the partial derivatives of \(L(\mathbf{P},\mathbf{B})\) with respect to \(p,u,\lambda,b\) are zero, leading to:  
\[ \begin{bmatrix}\phi^{T}\mathbf{D}_{\alpha}\phi+\mathbf{I}&\phi^{T}\mathbf{\alpha}\\ \mathbf{\alpha}^{T}\phi&\mathbf{I}^{T}\mathbf{\alpha}\end{bmatrix}\begin{bmatrix}w_{j}\\ b_{j}\end{bmatrix} = \begin{bmatrix}\phi^{T}\mathbf{D}_{\alpha}y_{j}\\ \mathbf{\alpha}^{T}y_{j}\end{bmatrix}  \tag{13}\]  
where \(\phi=[\phi(x_{1}),\cdots,\phi(x_{K})]^{T}\), \(\mathbf{D}_{\alpha}=diag(\alpha_{1},\cdots,\alpha_{K})\), \(\mathbf{\alpha}=[\alpha_{1},\cdots,\alpha_{K}]^{T}\), \(\mathbf{I}=[1,\cdots,1]^{T}\). Expressing \(w_{j}\) as a linear combination of samples in the feature space, i.e., \(w_{j}=\sum\phi(x_{i})\beta_{i}^{j}=\phi^{T}\beta^{j}\), and substituting into Equation (13), we obtain the following expression:  
\[ \begin{bmatrix}\mathbf{K}+\mathbf{D}_{\alpha}^{-1}&\mathbf{I}\\ \mathbf{\alpha}^{T}\mathbf{K}&\mathbf{I}^{T}\mathbf{\alpha}\end{bmatrix}\begin{bmatrix}\beta ^{j}\\ b_{j}\end{bmatrix} = \begin{bmatrix}y_{j}\\ \mathbf{\alpha}^{T}y_{j}\end{bmatrix}