Kriged road-traffic maps

Abstract

A common difficult problem of large cities with heavy traffic is the prediction of traffic jams. In this paper, a first step towards mathematical traffic forecasting, namely the spatial reconstruction of the present traffic state from pointwise measurements is briefly described. For details, we refer to [1], where models of stochastic geometry and geostatistics are used to spatially represent the traffic state by means of velocity maps. A corresponding Java software that implements efficient algorithms of spatial extrapolation is developed; see [5]. To illustrate our extrapolation method, we use real traffic data originating from downtown Berlin. It was provided to us by the Institute of Transport Research of the German Aerospace Center (DLR). Approximately 300 test vehicles (taxis) equipped with GPS sensors transmit their geographic coordinates and velocities to a central station within regular time intervals from 30s up to 6 min; see Fig. 2. Thus, a large data base of more than 13 million positions was formed since April 2001; In the first stage of our research, only a smaller data set (taxi positions on all working days from 30.09.2001 till 19.02.2002, 5.00-5.30 pm, moving taxis only) was considered. Furthermore, the observation window was reduced to downtown Berlin to avoid inhomogeneities in the taxi positions. The main idea of the extrapolation technique described in Sects. 2 and 3 below is to interpret the velocities of all vehicles at given time t as a realization of a spatial random field V (t) = {V (t, u)} where V (t, u) is a traffic velocity vector at location u R and time instant t 0. The goal is to analyze the spatial structure of these random fields of velocities in order to describe the geometry of traffic jams. Since V (t, u) can be measured just pointwise at some observation points u1,..., un, a spatial extrapolation of the observed data is necessary. Notice that the velocities strongly depend on the location and the direction of movement, e.g. the speed limits and consequently the mean velocities are higher on highways than in downtown streets. The classical extrapolation methods of geostatistics (see e.g. [6]) either make no use of additional directional information or provide measurements V (t, u + ui) and V (t, u ui) with equal weights. Both these features are not adequate to the setting mentioned above. Thus, the standard extrapolation methods had to be adapted to our specific problem. In Sect. 3, an ordinary kriging with moving neighborhood is described that allows to extrapolate directed velocity fields. First, the original data set should be split into four subsets which are directionally homogeneous. A data unit (u, V (t, u)) belongs to the data set i (i = 1, . . . , 4) if the polar angle of the vector V (t, u) lies within the directional sector Si = [0, 2) : (i 1)/2