Entropy metrics (for example, permutation entropy) are nonlinear measures of irregularity in time series (one-dimensional data). Some of these entropy metrics can be generalised to data on periodic structures such as a grid or lattice pattern (two-dimensional data) using its symmetry, thus enabling their application to images. However, these metrics have not been developed for signals sampled on irregular domains, defined by a graph. Here, we define for the first time an entropy metric to analyse signals measured over irregular graphs by generalising permutation entropy, a well-established nonlinear metric based on the comparison of neighbouring values within patterns in a time series. Our algorithm is based on comparing signal values on neighbouring nodes, using the adjacency matrix. We show that this generalisation preserves the properties of classical permutation for time series and the recent permutation entropy for images, and it can be applied to any graph structure with synthetic and real signals. We expect the present work to enable the extension of other nonlinear dynamic approaches to graph signals.

中文翻译：

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图信号的排列熵


熵度量（例如排列熵）是时间序列（一维数据）中不规则性的非线性度量。其中一些熵度量可以利用其对称性推广到周期性结构的数据，例如网格或晶格图案（二维数据），从而使其能够应用于图像。然而，这些指标尚未针对在由图表定义的不规则域上采样的信号而开发。在这里，我们首次定义了一种熵度量，通过推广排列熵来分析在不规则图上测量的信号，排列熵是一种基于时间序列模式内相邻值比较的成熟非线性度量。我们的算法基于使用邻接矩阵比较相邻节点上的信号值。我们证明这种推广保留了时间序列的经典排列和图像的最近排列熵的属性，并且它可以应用于具有合成和真实信号的任何图结构。我们期望目前的工作能够将其他非线性动态方法扩展到图形信号。