Many signals evolve in time as a stochastic process, randomly switching between states over discretely sampled time points. Here we make an explicit link between the underlying stochastic process of a signal that can take on a bounded continuum of values and a random walk process on a graphon. Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs whose size tends to infinity. We introduce transfer operators, such as the Koopman and Perron--Frobenius operators, associated with random walk processes on graphons and then illustrate how these operators can be estimated from signal data and how their eigenvalues and eigenfunctions can be used for detecting clusters, thereby extending conventional spectral clustering methods from graphs to graphons. Furthermore, we show that it is also possible to reconstruct transition probability densities and, if the random walk process is reversible, the graphon itself using only the signal. The resulting data-driven methods are applied to a variety of synthetic and real-world signals, including daily average temperatures and stock index values.

翻译：许多信号随时间以随机过程演化，在离散采样时间点上随机切换状态。本文建立了可取值于有界连续区间的信号底层随机过程与图极限上的随机游走过程之间的显式关联。图极限是表征规模趋于无穷的图序列收敛极限的无限维对象。我们引入与图极限上随机游走过程相关的转移算子（如Koopman算子和Perron–Frobenius算子），进而阐明如何从信号数据中估计这些算子，并利用其特征值与特征函数检测聚类，从而将传统谱聚类方法从图扩展至图极限。此外，我们证明仅利用信号本身即可重建转移概率密度，并在随机游走过程可逆的条件下重建图极限本身。所提出的数据驱动方法被应用于多种合成与真实世界信号，包括日均气温与股票指数数值。