## 2 Mobile Fingerprinting Based on Semisupervised Learning
### Semisupervised Pseudolabeling  
Here, the semisupervised optimization to create accurate pseudolabels for the unlabeled data by considering both the temporal and spatial aspect is presented. Given the labeled and unlabeled training data \(\{(y_{i},x_{i})\}_{i=1}^{K}\) arranged in chronological order, the optimization for generating pseudolabels \(\tilde{X}\in R^{K}\) is  
\[\min_{\tilde{X}\in R^{n}} \frac{c}{2}E^{\mathrm{T}}E+\frac{1}{2}\mu_{1}\tilde{X}^{\mathrm{T }}L\tilde{X}+\frac{1}{2}\mu_{2}\tilde{X}^{\mathrm{T}}DD^{\mathrm{T}}\tilde{X}\] (4) subject to \[:\Lambda\tilde{X}-X=E,\]  
where \(\Lambda\) is a diagonal matrix of trade-off parameters with \(\Lambda_{ii}=1\) if \(y_{i}\) is a labeled data point and \(\Lambda_{ii}=0\) if \(y_{i}\) is an unlabeled data point. \(\mu_{1}\) and \(\mu_{2}\) represent a trade-off relationship between spatial and temporal correlation, matrix \(D\) is defined in Equation (3), and \(X\in R^{K}\) is given by  
\[X=\left\{ \begin{array}{rl}&x_{i}&\text{if $y_{i}$ labeled}\\ &0&\text{else if $y_{i}$ unlabeled}\end{array} \right.,\]  
and graph Laplacian \(L\) is defined as \(L=B^{-1/2}(B-C)B^{-1/2}\), with the adjacency matrix \(C\) and the diagonal matrix \(B\) given by \(B_{ii}=\sum_{j=1}^{K}C_{ij}\). In general, the edge weights \(C_{ij}\) are defined as a Gaussian function of Euclidean distance, given by  
\[C_{ij}=\exp\left(-\|x_{i}-x_{j}\|^{2}/\sigma_{w}^{2}\right),\]  
where \(\sigma_{w}^{2}\) is the kernel width parameter. With the introduction of a multiplier \(\alpha_{P}\), the Lagrangian of Equation (4) is given by  
\[L=\frac{c}{2}E^{\mathrm{T}}E+\frac{1}{2}\mu_{1}\tilde{X}^{\mathrm{T}}L\tilde{ X}+\frac{1}{2}\mu_{2}\tilde{X}^{\mathrm{T}}DD^{\mathrm{T}}\tilde{X}+\alpha_{P} \left(\Lambda\tilde{X}-X-E\right) \tag{5}\]  
The derivatives of Equation (5) with respect to the variables \(\tilde{X}\), \(\alpha_{P}\), and \(e\) set to zero are  
\[\frac{\partial L}{\partial\tilde{X}} = \mu_{1}L\tilde{X}+\mu_{2}DD^{\mathrm{T}}\tilde{X}+\Lambda\alpha_{ P}=0 \tag{6}\] \[\frac{\partial L}{\partial e} = cE-\alpha_{P}=0\] (7) \[\frac{\partial L}{\partial\alpha_{P}} = \Lambda\tilde{X}-X-E=0 \tag{8}\]Substituting Equation (7) into Equation (8) gives  
\[\Lambda\tilde{X}-\frac{1}{c}\alpha_{P}=X. \tag{9}\]  
Then, the linear algebraic equations satisfying the KKT conditions are defined as follows,  
\[\Phi A=F, \tag{10}\]  
where  
\[\Phi=\left[ \begin{array}{cc}\mu_{1}L+\mu_{2}DD^{\mathrm{T}}&\Lambda\\ \Lambda&-\frac{1}{c}\end{array} \right] \tag{11}\]  
and,  
\[A=\left[ \begin{array}{c}\tilde{X}\\ \alpha_{P}\end{array} \right],\;\;F=\left[ \begin{array}{c}0\\ X\end{array} \right] \tag{12}\]  
Therefore, the optimal pseudolabels are obtained by inverse calculation of the matrix \(\Phi\). The advantages of this optimization compared to traditional semisupervised learning are as follows. * A closed-form solution can be obtained from the linear algebraic Equation (10), which is faster than for other semisupervised algorithms and requires an iterative quadratic programming (QP) solver. * The solution for \(\tilde{X}\) also inherits the sparsity characteristics of a support vector machine [42]. This is beneficial when we manage training data storage by saving reliable training data only, that is, large values of \(\alpha_{P}\) i.e., the support vectors. * In addition to spatial representation, a time-series representation is considered, where the relevant term, \(\mu_{2}DD^{\mathrm{T}}\), is inserted independently into the optimization problem. Without a substantial increase in computational time, it improves the accuracy of the pseudolabels. Its performance is analyzed in the next section. In a summary, we address the role of the mobile fingerprinting and the connection to the next sections. The mobile fingerprinting is a method for database construction. Especially, it is useful when amount of the labeled data is not enough by its capability of accurately labelling the positions of the unlabeled data.