# A Data-Driven Deep Learning Framework for Prediction of Traffic Crashes at Road Intersections
## 3 Preliminaries
### Problem Formulation  
**Definition 1**. : _Road network. A road network is denoted as a directed graph \(\mathcal{G}=(V,E,\Psi,\Omega)\), where \(V\) is a set of nodes, i.e., intersections (without loss of generality, we also consider terminal/end of a road as an intersection); \(E\subseteq V\times V\) is a set of directed edges that denotes road segments; \(\Psi:V\rightarrow\mathbb{A}\) is a descriptor function that describes a node \(v\in V\) by a set of pre-defined tags in \(\mathbb{A}\) to capture its geospatial characteristics, such as intersection tags and intersection types, i.e., \(\Psi(v)\in\mathbb{A}\) and \(\Omega:E\rightarrow\mathbb{B}\) is a descriptor function that describes an edge \(e\in E\) by a set of pre-defined tags in \(\mathbb{B}\) to capture its geospatial characteristics, i.e., \(\Omega(e)\in\mathbb{B}\), such as road segment categories, and the geo-locality relationship between intersections and road segments, etc._  
**Definition 2**. : _Traffic Crash Severity. For each traffic crash \(j\), we define the traffic crash severity as follows:_  
\[\text{Severity}_{j}=\omega_{I}(n_{\text{ped}_{i}}+n_{\text{ped}_{i}})+\omega_{K} (n_{\text{ped}_{k}}+n_{\text{ped}_{k}}) \tag{1}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Datasets** & **New York City (1 January 2014 to 30 June 2016)** \\ \hline Road Network & 389,312 intersections and 513,099 road segments \\ Traffic Crashes & 654 K crash records \\ POIs Data & 20,469 POIs from 13 categories \\ Taxi Trips & 244,199K GPS trajectories of taxi cabs \\ Hourly weather & 21,840 weather records \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistics of the datasets. Figure 3: The Relationship between Traffic Crashes and Weather Types at Road Intersections from 2014 to 2016.  
where \(n_{\text{ped}_{i}}\) or \(n_{\text{ped}_{i}}\) are the number of pedestrians and vehicle occupants injured, \(n_{\text{ped}_{k}}\) and \(n_{\text{ped}_{k}}\) are the number of pedestrians and vehicle occupants killed, \(\omega_{I}\) and \(\omega_{K}\) are the weight of injured number and killed number, which are hyper-parameters._  
According to a report from the U.S. Transportation Security Administration, the severity of traffic crashes is divided into four levels, with 1 being the least and 4 being the most serious. Thus, we classify all crashes into four crash categories [22] by \(Severity_{i}\), i.e., \(\mathcal{L}=\{0,\,1,\,2,\,\geq 3\}\), i.e., zero injured crashes, minor injured crashes, injured crashes and fatal crashes to the four levels. Definition 3: Intersection Hazard Level. According to the number of casualties in traffic crashes, we define the intersection hazard level in time interval \(\Delta t\), denoted as \(\text{Hazard}_{\text{v}_{i}}(\Delta t)\), to describe how dangerous an intersection is. It is denoted as the summation of all the crash categories of crashes that occurred at the intersection \(v_{i}\) within time interval \(\Delta t\). Definition 4: Crash Features. The historical traffic crash records contains rich information and frequency correlation to traffic crashes. We extract the crash features from historical traffic crash records as follows:  
\[\mathcal{C}_{\text{v}_{i}}=\{\text{Hazard}_{\text{v}_{i}}(\Delta t),\text{IC} _{\text{v}_{i}}(\Delta t),\text{TC}_{\text{v}_{i}}\} \tag{2}\]  
where \(\text{IC}_{\text{v}_{i}}(\Delta t)\) denotes the number of traffic crashes that occurred at the intersection \(v_{i}\) in a time interval \(\Delta t\), and \(\text{TC}_{\text{v}_{i}}\) is the total number of traffic crashes at the intersection \(v_{i}\) during the whole collection period of the historical records. Definition 5: Additional Thematic Information (ATI). In addition to capturing topological features, we consider four types of ATI, including the traffic situation, weather, and the environmental and calendar features associated with intersections.