The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points, such as the total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of the effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator by an application of a Berezin-Lieb inequality. The framework for our approach is provided by quantum harmonic analysis.

翻译：本文旨在开发理解函数系统（例如时间序列数据点）局部结构的新工具，包括总相关函数、数据集的Cohen类、数据算子以及平均集中度缺失。数据算子的Cohen类为数据集提供了时频表示。此外，我们证明了数据算子的冯·诺依曼熵能够捕捉数据集的局部特征，并且与有效维度的概念相关。数据算子的累积Cohen类为数据集提供了低维表示，我们通过应用Berezin-Lieb不等式，利用平均集中度缺失和数据算子的冯·诺依曼熵来量化这一表示。本文的方法框架由量子调和分析提供支撑。