# Fingerprinting Indoor Positioning Based on Improved Sequential Deep Learning
## 3 System Overview
### Region Estimation  
Consider the following general nonparametric model for the RSSI readings: for each observation \(i=1,\cdots,N\), the model is represented as follows:  
\[S_{i}=m(L_{i},\epsilon_{i},\eta), \tag{5}\]  
where \(S_{i}\in\mathbf{R}^{R}\) is \(R\) dimensional RSSI measurements, it is similar to (1). However, in the method proposed in this article, the values measured at the same location are treated as separate data points rather than being averaged in (2). Additionally, we use \(m_{r}\) to represent the \(r\)th component of \(m\), which connects the RSSI measurement from the \(r\)th AP to its corresponding structural inputs. \[S_{ir}=m_{r}(L_{i},\epsilon_{i},\eta_{r}). \tag{6}\]  
Moreover, based on the marginal distribution of the structural inputs \((L_{i},\epsilon_{i},\eta_{r})\), \(\mathbb{E}(L_{i}|S_{i})\) can be derived using Bayes' rule. We define \(h_{0}(\cdot)\equiv\mathbb{E}(L_{i}|S_{i}=\cdot)\) as the infinite-dimensional function of interest to be estimated. Unlike traditional machine learning localization approaches, which directly estimate \(h_{0}(\cdot)\) from large training samples and use the approximation \(\hat{h}(S_{i})\) to predict \(L_{i}\), the sequential learning algorithm, as outlined in Steps 1-4 in Section 3.1, offers a more reliable solution. First, partition the location space \(L\) into two regions, \(\mathcal{L}_{0}\) and \(\mathcal{L}_{1}\), using a simple hyperplane, such that \(\mathcal{L}=\mathcal{L}_{0}\cap\mathcal{L}_{1}\) and \(\mathcal{L}_{0}\cup\mathcal{L}_{1}=\emptyset\). Define a dummy variable \(Z_{i}=1(L_{i}\in\mathcal{L}_{1})\), where \(Z_{i}=z\) indicates that location \(L_{i}\) is within \(\mathcal{L}_{z}\). While we use a binary partition here, \(\mathcal{L}\) could be divided into multiple categories \((K\geq 2)\). By utilizing the feature \(Z_{i}\), we apply a binary classification algorithm to infer \(Z_{i}\) from \(S_{i}\), which is a simpler task compared to the original localization problem. By selecting an appropriate partition, we aim for near 100% accuracy. Next, define our target function as \(h_{0}^{*}(S_{i},z)\equiv\mathbb{E}(L_{i}|S_{i},Z_{i}=z)\) for \(z=0,1\), and estimate \(h_{0}^{*}(\cdot,z)\) using machine learning. Notably, we focus on subsamples with \(Z_{i}=z\) during training, rather than using the entire dataset. Finally, we divide the training data \(\mathcal{T}\) into \(N_{F}\) regions, denoted as \(\mathcal{L}_{\mathcal{F}_{i}}\), where \(j=1,\cdots,N_{F}\). Figure 1: The whole architecture of the proposed method. **Remark 1**: _These regions could represent a building or a specific office on a particular floor within a building._  
This approach offers two significant advantages. First, the relationship captured by \(h_{0}^{*}(\cdot,z)\) is inherently simpler and more straightforward for model selection, as it focuses on how \(L_{i}\) depends on \(S_{i}\) within the specific context of \(Z_{i}=z\). This makes the estimation process more manageable compared to the broader \(h_{0}(\cdot)\) relationship. Second, by training on subsamples where \(Z_{i}=z\), the method effectively reduces noise from locations outside \(L_{z}\). This localized approach is especially effective in addressing observational-level heterogeneity caused by mobile devices, resulting in more reliable outcomes. **Remark 2**: _The classification algorithm is required to achieve quite perfect accuracy. If no binary partition meets this condition, the partitioning of the location space should be halted. This requirement indicates that the variable \(Z_{i}\), as a binary feature of \(L_{i}\), should have a deterministic relationship with \(S_{i}\). By keeping the binary feature variable \(Z_{i}\) fixed, we can control the high-dimensional \(S_{i}\) within a localized area._  
1. Input: Observations \(\{(S_{i},L_{i})\}_{i\in\mathcal{T}}\) and \(\{(S_{i},L_{i})\}_{i\in\mathcal{V}}\). 2. Initialization: Start with an unlabeled root node associated with \(L_{0}\). 3. While there is an unlabeled leaf \(\mathcal{L_{F}}\), do the following:  
1.