Local Structure and effective Dimensionality of Time Series Data Sets

TL;DR

This paper introduces new tools based on quantum harmonic analysis to analyze the local structure and effective dimensionality of time series data, using concepts like the Cohen class, data operator, and von Neumann entropy.

Contribution

It develops novel methods for understanding local features and dimensionality of time series data through quantum harmonic analysis and related operators.

Findings

The von Neumann entropy of the data operator captures local data features.

The Cohen class provides a time-frequency representation of data sets.

The accumulated Cohen class yields a low-dimensional data representation.

Abstract

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.