58 research outputs found
Linear processes in high-dimension: phase space and critical properties
In this work we investigate the generic properties of a stochastic linear
model in the regime of high-dimensionality. We consider in particular the
Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We
analyze both deterministic and random versions of these models, showing the
existence of a stable and an unstable phase. We find that along the transition
region separating the two regimes, the correlations of the process decay
slowly, and we characterize the conditions under which these slow correlations
are expected to become power-laws. We check our findings with numerical
simulations showing remarkable agreement with our predictions. We finally argue
that real systems with a strong degree of self-interaction are naturally
characterized by this type of slow relaxation of the correlations.Comment: 40 pages, 5 figure
Uncovering Causality from Multivariate Hawkes Integrated Cumulants
We design a new nonparametric method that allows one to estimate the matrix
of integrated kernels of a multivariate Hawkes process. This matrix not only
encodes the mutual influences of each nodes of the process, but also
disentangles the causality relationships between them. Our approach is the
first that leads to an estimation of this matrix without any parametric
modeling and estimation of the kernels themselves. A consequence is that it can
give an estimation of causality relationships between nodes (or users), based
on their activity timestamps (on a social network for instance), without
knowing or estimating the shape of the activities lifetime. For that purpose,
we introduce a moment matching method that fits the third-order integrated
cumulants of the process. We show on numerical experiments that our approach is
indeed very robust to the shape of the kernels, and gives appealing results on
the MemeTracker database
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