Stationary Processes on Directed Graphs

In this paper, we extend the concepts of kernels, weak stationarity and white noise to the case of directed graphs (digraphs). We shall determine the attributes of two types of kernels (I and II) based on the Jordan decomposition of the shift operator. Using kernels of type I and II, we define the stationarity concept and filtering properties on digraphs. We further generalize the concept of Wiener filter and the related optimization framework to digraphs. To validate the concepts, we apply our theories numerically to denoise and recover temperature values in some cities in France based on real recordings that are manually noise contaminated.