Local stationarity of graph signals: insights and experiments

Abstract

In this paper, we look at one of the most crucial ingredient to graph signal processing: the graph. By taking a step back on the conventional approach using Gaussian weights, we are able to obtain a better spectral representation of a stochastic graph signal. Our approach focuses on learning the weights of the graphs, thus enabling better richness in the structure by incorporating both the distance and the local structure into the weights. Our results show that the graph power spectrum we obtain is closer to what we expect, and stationarity is better preserved when going from a continuous signal to its sampled counterpart on the graph. We further validate the approach on a real weather dataset.

A Joint Markov Model for Communities, Connectivity and Signals Defined Over Graphs

“…, which leads to the so-called Gaussian weights [16], i.e., a ij = exp (−||xi−xj|| 2 ) . Thus, the CMRF extends to real weights, and the graph learning in (5) includes as particular cases Algorithm 1 : Joint Signal and Graph Topology Recovery Data: y, γ.…”

“…Gaussian distance weights [16], covariance based weights [22], Intensity/Distance based weights [14], among the others. Finally, we highlight that our learning strategy in (5) might also be modified to select edges whose potential energy falls below a given threshold θ.…”

“…Also, in [14], the authors infer the graph topology based on the potential energy of the associated MRF. In [16], SoG are modeled as an irregular sampling of a continuous space GMRF. Finally, in [17] the authors proposed an MRF to model edges and signals observed over graphs in a joint fashion.…”

Piecewise Stationary Modeling of Random Processes Over Graphs With an Application to Traffic Prediction

“…Piecewise stationary modeling has also been explored in time series analysis which is mainly achieved by detecting change points [20], [21]. Recently, there has been an effort in graph signal processing literature to detect non-stationary vertices (change points) in a random process over graph [22], [23], however, to achieve that, authors introduce another definition for stationarity, called local stationarity, which is different from the widely-used definition of GWSS.…”

Graph Variogram: A Novel Tool to Measure Spatial Stationarity

“…Recent contributions to extending stationarity to graph signals have considered both global [5][6][7][8] and local [9,10] definitions of graph stationarity. The former is summarized as second order graph stationarity characterized by uncorrelated spectral components leading to a straightforward definition of graph PSD [6].…”