Directional Assessment of Traffic Flow Extremes

Abstract

We analyze extremes of traffic flow profiles composed of traffic counts over a day. The data is essentially curves and determining which trajectory should be classified as extreme is not straight forward. To assess the extremes of the traffic flow curves in a coherent way, we use a directional definition of extremeness and apply the dimension reduction technique called principal component analysis (PCA) in an asymmetric norm. In the classical PCA one reduces the dimensions of the data by projecting it in the direction of the largest variation of the projection around its mean. In the PCA in an asymmetric norm one chooses the projection directions, such that the asymmetrically weighted variation around a tail index -- an expectile -- of the data is the largest possible. Expectiles are tail measures that generalize the mean in a similar manner as quantiles generalize the median. Focusing on the asymmetrically weighted variation around an expectile of the data, we find the appropriate projection directions and the low dimensional representation of the traffic flow profiles that uncover different patterns in their extremes. Using the traffic flow data from the roundabout on Ernst-Reuter-Platz in the city center of Berlin, Germany, we estimate, visualize and interpret the resulting principal expectile components. The corresponding directional extremes of the traffic flow profiles are simple to identify and to connect to their location- and time-related specifics. Their shapes are driven by their scores on each principal expectile component which is useful for extracting and analyzing traffic patterns. Our approach to dimensionality reduction towards the directional extremes of traffic flow extends the related methodological basis and gives promising results for subsequent analysis, prediction and control of traffic flow patterns.