# Predicting Human Location Using Correlated Movements
## 6 The PCM-Based Location Prediction (PLP) Model  
In this section, we propose the PCM-based location prediction (PLP) model based on the neural network to predict human mobility given the movement patterns of \(r\) selected PCMs. Figure 4 illustrates the NN-based prediction model where the location of a person at time \(t\) is estimated given the information of \(r\) PCMs, i.e., \(z_{i_{1}}^{i_{2}}=\{z_{i,T}^{\text{loc}},z_{i}^{\text{day}},z_{i}^{\text{time}}\}\). Spatial information \(z_{i,T}^{\text{loc}}\), where \(1\leq l\leq r\) and \(i_{1}\leq\tau\leq i_{2}\), indicates the symbolic location of the \(r\) PCMs from time \(i_{1}\) to \(i_{2}\). Meanwhile, temporal information \(z_{i}^{\text{day}}\) and \(z_{i}^{\text{time}}\) are the indices of the day of the week and the time slot in the day at time \(t\), respectively. Let \(y_{\text{NN}}^{\text{pred}}\triangleq f_{\text{NN}}(z_{i,T}^{\text{loc}},z_{ i}^{\text{day}},z_{i}^{\text{time}})\) denote the output of the NN-based predictor, i.e., the most likely location of the person during time slot \(t\). Note that to predict the location at time \(t\) of \(m\) persons in the community, \(m\) corresponding NN-based prediction models need to be trained separately. The remainder of this subsection describes an NN-based predictor. The function \(f_{\text{NN}}(z_{i,T}^{\text{loc}},z_{i}^{\text{day}},z_{i}^{\text{time}})\) can be decomposed into three parts, as follows. Figure 4: The PCM-based location prediction model. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(m\)} & \(I_{\text{BSB}}\) & \(I_{\text{IFS}}\) & \(I_{\text{FS}}\) \\  & **(millions)** & **(millions)** & **(millions)** \\ \hline
43 & 7.00 & 59.02 & 158.30 \\
100 & 15.36 & 142.09 & 878.17 \\
500 & 74.06 & 725.03 & 2229.0 \\
1000 & 147.42 & 1453.7 & 8933.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the number of parameters among behavioral similarity-based (BSB), ignored feature selection (IFS), and forward search (FS) methods. 1. The contextual information of \(r\) PCMs, i.e., \(z^{\text{loc}}_{1,i_{1}}z^{\text{loc}}_{2,1j_{2}},...,z^{\text{loc}}_{r,j_{2}}z^{ \text{day}}_{t},z^{\text{time}}_{t}\), is mapped onto real vectors using the one-hot encoding function \(c(. )\), where output is the indicator vector in which only one element is set to 1 while others are 0. Since \(|\mathbb{L}|\) is the size of the extracted location set, \(c(z^{\text{loc}}_{1,i_{1}}),c(z^{\text{loc}}_{1,i_{2}}),...,c(z^{\text{loc}}_{ r,i_{2}})\) are vectors in \(\mathbb{B}^{|\mathbb{L}|}\), where \(\mathbb{B}\) is a set of binary numbers; \(c(z^{\text{day}}_{t})\) is a day-index vector in \(\mathbb{B}^{7}\). Meanwhile, the cellular trace between 8 a.m. and 12 p.m. is extracted and each time slot lasts 30 min; therefore, \(z^{\text{time}}_{t}\) is mapped onto a 32-dimensional binary vector. In addition, grouped time slots are considered. In this case, \(c(z^{\text{time}}_{t})\) is a three-dimensional vector where each dimension corresponds to one of the three parts of a day, i.e., morning, afternoon, and evening. 2. The NN classifier maps input sequence \(c(z^{i_{2}}_{i_{1}})=\{c(z^{\text{loc}}_{i,\tau}),c(z^{\text{day}}_{t}),c(z^{ \text{time}}_{t})\}\) to a conditional probability distribution over locations. Let \(\mathbf{v}^{(i)}\) denote an indicator vector in which only the \(i^{\text{th}}\) element is 1 and others are 0 (\(1\leq i\leq|\mathbb{L}|\)). Vector \(\mathbf{t}\) is defined as the target vector of the NN classifier. The output of the NN is denoted by vector \(\hat{\mathbf{y}}_{\text{NN}}\) where the \(i^{\text{th}}\) element, \(\hat{y}_{i}\), estimates posterior probability \(\hat{y}_{i}=P(\mathbf{t}=\mathbf{v}^{(i)}|c(z^{i_{2}}_{i_{1}}))\) that the person stays in location \(i\) given the information of \(r\) PCMs. 3.