# Forecasting Taxi Demands Using Generative Adversarial Networks with Multi-Source Data
## 3 Problem Definition  
In this section, firstly we define several key concepts, and then formally formulate the taxi demand forecasting problem.  
Definition 1 (_taxi trip_). A taxi trip is defined as a tuple (ID, \(\text{pickup}_{\text{Ptime}}\), \(\text{pick}_{\text{location}}\),\(\text{drop}_{\text{time}}\), \(\text{drop}_{\text{location}}\), duration, distance, fare), where ID represents the trip identification, \(\text{pickup}_{\text{time}}\) is the pick-up time (in hours and minutes), \(\text{pick}_{\text{location}}\) is the pick-up location (longitude and latitude), \(\text{drop}_{\text{time}}\) is the drop-off time, \(\text{drop}_{\text{location}}\) represents the drop-off location, duration, distance, and fare are the calculated duration, distance and, fare of a trip.  
Definition 2 (_Road Network_). A road network of a city is composed of a set of road segments. Each road segment is associated with two terminal points (i.e., intersections of crossroads), and connects with other road segments by sharing the same terminals. All road segments compose the road network in the format of a graph.  
Definition 3 (_Temporal Data_). For the sake of a detailed description of the spatial and temporal dimensions of fine-grained taxi demands, the time is discretized into time slots \(t\) (date, hour, and minutes).  
Definition 4 (_Point of Interest POI_). POI represented a place (like a restaurant) in an urban area r. Each POI \(\text{p}_{i}\) associated with a location \(p_{i}.l\) and a POI category \(p_{i}.c\in C\), where C is a set of categories.  
Definition 5 (_Weather_). Weather presents several parameters related to weather status in a determined time slot (date, hour, and minutes) of an urban area r. Each weather may consider several associated parameters including temperature, humidity, weather status (rain, sunny), wind, etc.  
Here, we can provide a formal definition of fine-grained taxi demands as follows.  
Definition 6 (_Taxi Demands_). We use \(\text{Y}_{i}\) to represent the amount of taxi pick-ups (demand of taxis) in an area \(r\in R\) at a time slot \(t\in T\). Hence, taxi demands in a time slot \(t\) are defined as \(\text{Y}_{r,t}=[\text{Y}_{0,t},\text{Y}_{1,t},\ldots,\text{Y}_{n,t}]\).  
To forecast taxi demands in future, we first dig fine-grained taxi demands in past time slots from historical taxi trips data defined as in Definition 1 As Taxi Demands have a strong correlation with other temporal, spatial and external factors, we collect significant factors related to taxi demands (as in Definitions 2,3,4 and 5) in the determined area \(r\), at time slot \(t\).  
Now, we can define the problem of forecasting the fine-grained taxi demands in a determined time slot \(t\) as follows.  
Definition 7 (_Taxi Demand Forecasting Problem_). Consider an urban area is divided into disjoint regions \(R\) by the road network.  
Given fine-grained taxi demands in the past time slots \(\{\text{Y}_{t}\,|\,_{t}=0,\,1,\,\ldots,t\}\) extracted from historical taxi trips data (pick-ups), we try to predict fine-grained taxi demands at a determined time slot. The taxi demand forecasting is denoted as \(\widehat{\text{Y}}_{t+1}=[\text{\ }\widehat{\text{Y}}_{1,t+1},\ldots,\, \widehat{\text{Y}}_{n,t+1}]\).