# Predicting Human Location Using Correlated Movements
## 4 The Proposed Mobility Prediction Model
### Two-Phase Mobility Prediction Framework  
The proposed human mobility prediction framework consists of two phases, i.e., detection of persons with correlated movements (PCMs) and PCM-based location prediction, as shown in Figure 1. Let \(r\) denote the number of selected PCMs. In the first phase, \(r\) from among \((m-1)\) remaining members are extracted as PCMs of person \(p\). Then, given the location information of the chosen members, the current or future position of person \(p\) is predicted. In the first phase, we measure independently the social correlation between person \(p\) and each of the \((m-1)\) other members. In order to choose \(r\) useful PCMs for person \(p\), these \((m-1)\) people are arranged according to their correlation scores, and \(r\) members with the highest scores are selected. The training set is used in the first phase where the input is vector \(\mathbf{x}\) that represents positional information of the \((m-1)\) remaining members. Table 2 shows an example of vector \(\mathbf{x}\) when the location of an arbitrary person, \(p\), is estimated, where \(\mathbf{x}\) consists of spatial and temporal information. More specifically, \(x_{1}^{\text{loc}},x_{2}^{\text{loc}},...,x_{p-1}^{\text{loc}}\), \(x_{p+1}^{\text{loc}}\),..., \(x_{m}^{\text{loc}}\) denotes the positions of the \((m-1)\) remaining members, while \(x^{\text{day}}\) and \(x^{\text{time}}\) account for the day and time slot indices, respectively. The first phase returns \(r\) PCMs of person \(p\) as the output. Then, the location prediction phase estimates the most likely current or future position of person \(p\) using the spatio-temporal information of the \(r\) selected PCMs. Each sample of the training set is given by \((\mathbf{z}_{i_{1}}^{i_{2}},\mathbf{y}_{i})\), where label \(\mathbf{y}_{i}\) is location information of person \(p\) at time \(t\), and feature vector \(\mathbf{z}_{i_{1}}^{i_{2}}\) is spatio-temporal information of PCMs from time slots \(i_{1}\) to \(i_{2}\) where \(1\leq i_{1}\leq i_{2}\leq t\). In the proposed framework, a conditional probability distribution is estimated with the objective of maximizing probability \(P(\mathbf{y}_{t}|\mathbf{z}_{i_{1}}^{i_{2}})\). More specifically, given the location information of PCMs at time slots between \(i_{1}\) and \(i_{2}\), the position of person \(p\) at time \(t\) needs to be predicted. In this work, \((t-1)\leq i_{1}\leq i_{2}\leq t\) is considered, i.e., \((i_{1},i_{2})=\{(t-1,t-1),(t-1,t),(t,t)\}\). In cases where \(i_{1}=i_{2}=(t-1)\), the location of PCMs in the previous time slot \((t-1)\) is used to predict the next location of person \(p\) (i.e., estimate person \(p\)'s future location at time \(t\)). Meanwhile, \((i_{1},i_{2})=(t-1,t)\) indicates that the model estimates the current position of the person using previous and current location information of PCMs. Whereas, if \(i_{1}=i_{2}=t\), the current position of the person is estimated given the information of PCMs at time \(t\). Hereafter, the model is regarded as estimating the location of a person at time slot \(t\). In the second phase, the PCM-based location prediction model is used to label an unknown sample as one out of \(|\mathbb{L}|\) possible locations. Figure 1: The proposed framework for predicting the current or future location of person \(p\). There are several benefits from the proposed framework. Our model does not require a recent location sequence of person \(p\) when predicting the current or future location of person \(p\), which is beneficial in cases where the location information of that person is not available. Note, however, that the location information of person \(p\) is necessary for training parameters of the prediction model. The proposed framework first selects \(r\) PCMs based on human mobility characteristics, e.g., encounter frequency and behavioral patterns. Then, the location information of only these chosen PCMs is used to predict a person's location at time \(t\).