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. We utilize the double cross-validation for determining the optimal component number and forecast traffic flow profiles based on the estimated model. 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 the reflected patterns.

我们分析了由一天的交通计数组成的极端交通流剖面。数据本质上是曲线，确定哪个轨迹应该被归类为极端不是直接的。为了以一致的方式评估交通流曲线的极值，我们使用极值的方向定义，并在非对称范数中应用称为主成分分析 (PCA) 的降维技术。在经典的 PCA 中，通过将数据投影到围绕其均值的最大投影变化方向来减少数据的维度。在非对称范数的 PCA 中，选择投影方向，使得围绕数据的尾部索引（期望值）的非对称加权变化尽可能大。期望值是尾部度量，它以类似于分位数泛化中位数的方式泛化平均值。关注围绕数据期望值的非对称加权变化，我们找到了适当的投影方向和交通流剖面的低维表示，它们揭示了极端情况下的不同模式。使用来自德国柏林市中心 Ernst-Reuter-Platz 环岛的交通流量数据，我们估计、可视化和解释了由此产生的主要预期分量。交通流剖面的相应方向极值很容易识别并连接到它们的位置和时间相关的细节。它们的形状由它们在每个主要期望分量上的分数驱动，这对于提取和分析流量模式很有用。我们利用双重交叉验证来确定最佳组件数量并根据估计模型预测交通流剖面。我们针对交通流方向极值的降维方法扩展了相关的方法论基础，并为后续分析、预测和控制反射模式提供了有希望的结果。