Section: New Results

Graph & Signal Processing

Participants : Paulo Gonçalves Andrade, Éric Fleury, Benjamin Girault, Sarra Ben Alaya.

Isometric Graph shift operator.

In [14] , [40] , we proposed a new shift operator for graph signals, enforcing that our operator is isometric. Doing so, we ensure that as many properties of the time shift as possible get carried over. Finally, we show that our operator behaves reasonably for graph signals.

Stationary graph signals.

We extended the concept of stationary temporal signals to stationary graph signals [24] . We introduced the concept of strict sense stationary and wide sense stationary graph signals as a statistical invariance through an isometric graph translation. Using these definitions, we proposed a spectral characterisation of WSS graph signals allowing to study stationarity using only the spectral components of a graph signal. Finally, we applied this characterisation to a synthetic graph in order to study a few important stochastic graph signals. Also, using geographic data, we analysed data from a graph set of weather stations and showed evidence of stationarity in the temperature signal [36] .

Community mining with graph filters for correlation matrices.

Communities are an important type of structure in networks. Graph filters, such as wavelet filter-banks, have been used to detect such communities as groups of nodes more densely connected together than with the outsiders. When dealing with times series, it is possible to build a relational network based on the correlation matrix. However, in such a network, weights assigned to each edge have different properties than those of usual adjacency matrices. As a result, classical community detection methods based on modularity optimisation are not consistent and the modularity needs to be redefined to take into account the structure of the correlation from random matrix theory. In our contribution [34] , we addressed how to detect communities from correlation matrices, by filtering global modes and random parts using properties that are specific to the distribution of correlation eigenvalues. Based on a Louvain approach, an algorithm to detect multiscale communities was also developed, which yields a weighted hierarchy of communities. The implementation of the method using graph filters was also discussed.

A strong Tauberian theorem for characteristic functions.

In [20] , we showed that a characteristic function which can be approximated at 0 by any polynomial of order n is actually n-times differentiable at 0. This fact is exploited to strengthen a tauberian-type result by Lukacs and provides the theoretical basis for a wavelet based non-parametric estimator of the tail index of a distribution. This work is a technical improvement of our previous contribution [53] .

Fractal Analysis of Fetal Heart Rate Variability.

The fetal heart rate (FHR) is commonly monitored during labor to detect early fetal acidosis. FHR variability is traditionally investigated using Fourier transform, often with adult predefined frequency band powers and the corresponding LF/HF ratio. However, fetal conditions differ from adults and modify spectrum repartition along frequencies. The study we reported in [12] questioned the arbitrariness definition and relevance of the frequency band splitting procedure, and thus of the calculation of the underlying LF/HF ratio, as efficient tools for characterising intrapartum FHR variability. Then, we showed that the intrapartum FHR is characterised by fractal temporal dynamics and promotes the Hurst parameter as a potential marker of fetal acidosis. This parameter preserves the intuition of a power frequency balance, while avoiding the frequency band splitting procedure and thus the arbitrary choice of a frequency separating bands. The study also shows that extending the frequency range covered by the adult-based bands to higher and lower frequencies permits the Hurst parameter to achieve better performance for identifying fetal acidosis.