# A Quadrilateral Geometry Classification Method and Device for Femtocell Positioning Networks
## 3 Principle of NMLP Geometry Classifier  
The architecture of the NMLP geometry classifier is shown in Figure 2. It consists of input data normalization, data scaling, an input layer, two hidden layers, and an output layer. The input vectors include:  
\[ \begin{array}{l}s_{i0}^{\rightarrow}s_{i1}=(x_{i1}-x_{i0},y_{i1}-y_{i0},z_{i1} -z_{i0})\\ s_{i0}^{\rightarrow}s_{i2}=(x_{i2}-x_{i0},y_{i2}-y_{i0},z_{i2}-z_{i0})\\ s_{i0}^{\rightarrow}s_{i3}=(x_{i3}-x_{i0},y_{i3}-y_{i0},z_{i3}-z_{i0})\\ s_{i1}^{\rightarrow}s_{i3}=(x_{i3}-x_{i1},y_{i3}-y_{i1},z_{i3}-z_{i1})\end{array}  \tag{7}\]  
where the coordinates of \(P\) set of the four FeNBs are \(S_{i}=s_{i0}(x_{0},y_{0},z_{0})s_{i1}(x_{1},y_{1},z_{1})s_{i2}(x_{2},y_{2},z_{ 2})\)\(s_{i3}(x_{3},y_{3},z_{3})\); \(i~{}=~{}1,2,...,P\)., where \(s_{ij}\) is the \(j\)th component of the \(i\)th input vector and \(P\) is the total number of four FeNBs in the coverage range of the target MUE. Note that each component in a vector of a quadrilateral is constituted by four FeNBs. The data normalization is obtained by:  
\[ \begin{array}{l}{S_{i0}^{\rightarrow}}{S_{i1}^{\rightarrow}}^{\prime}= \frac{S_{i0}S_{i1}}{L_{i\max}}=((x_{i1}-x_{i0})^{\prime},(y_{i1}-y_{i0})^{ \prime},(z_{i1}-z_{i0})^{\prime})=(d_{i1},d_{i2},d_{i3})\\ {S_{i0}^{\rightarrow}}{S_{i2}^{\rightarrow}}^{\prime}=\frac{S_{i0}S_{i2}}{L_ {i\max}}=((x_{i2}-x_{i0})^{\prime},(y_{i2}-y_{i0})^{\prime},(z_{i2}-z_{i0})^{ \prime})=(d_{i4},d_{i5},d_{i6})\\ {S_{i0}^{\rightarrow}}{S_{i3}}=\frac{S_{i0}S_{i3}}{L_{i\max}}=((x_{i3}-x_{i0})^ {\prime},(y_{i3}-y_{i0})^{\prime},(z_{i3}-z_{i0})^{\prime})=(d_{i7},d_{i8},d_{ i9})\\ {S_{i1}^{\rightarrow}}{S_{i3}^{\rightarrow}}=\frac{S_{i1}S_{i3}}{L_{i\max}}=( (x_{i3}-x_{i1})^{\prime},(y_{i3}-y_{i1})^{\prime},(z_{i3}-z_{i1})^{\prime})=(d_ {i10},d_{i11},d_{i12})\end{array}  \tag{8}\]  
where \(L_{\max}=\text{maximum of}\{\|s_{i0}^{\rightarrow}s_{i1}\|,\|s_{i1}^{\rightarrow}s_{i2}\|,\|s_{i2}^{ \rightarrow}s_{i3}\|,\|s_{i3}^{\rightarrow}s_{i0}\|\}\). The scaled input and output data are required for the MLP, in order to provide the correct value for an activation function. The linear scaling of data is the transformation from the original data variability within the interval \([d_{q,\min},d_{q,\max}]\) to the interval [0.1, 0.99], and can be performed using the Formula [10]:  
\[\eta_{iq}=\frac{(d_{iq}-d_{q,\min})(b^{\prime}-a^{\prime})}{d_{q,\max}-d_{q, \min}},~{}q=1,...,Q;i=1,...,P \tag{9}\]  
where \([a^{\prime},b^{\prime}]=[0.1,0.99]\). \(d_{iq}\) is the original length of the \(i\)th input component at the \(q\)th neuron of the input layer. The MLP network of the geometry classifier is composed of 12 (Q) inputs, which represent 12 features of the NMLP geometry classifier. The sigmoid activation function [12] used is non-linear, differentiable, and is defined by:  
\[y(a_{n}(k))=1/\left(1+e^{-a_{n}(k)}\right) \tag{10}\]  
Figure 2: Architecture of the NMLP geometry classifier. the neuron outputs of last layer where \(m_{l}=5\) is the total number of neurons in the last layer. Given the scaled training data: \(\mathbf{D}=\{(S_{i1}(k),D_{i1}(k)),...,\,(S_{im_{l}}(k),D_{im_{l}}(k))\}\) where \(S_{iq}(k),D_{iq}(k)\) are the given input and the desired output at the \(k\)th iteration.