21 research outputs found
Nonparametric Markovian Learning of Triggering Kernels for Mutually Exciting and Mutually Inhibiting Multivariate Hawkes Processes
In this paper, we address the problem of fitting multivariate Hawkes
processes to potentially large-scale data in a setting where series of events
are not only mutually-exciting but can also exhibit inhibitive patterns. We
focus on nonparametric learning and propose a novel algorithm called MEMIP
(Markovian Estimation of Mutually Interacting Processes) that makes use of
polynomial approximation theory and self-concordant analysis in order to learn
both triggering kernels and base intensities of events. Moreover, considering
that N historical observations are available, the algorithm performs
log-likelihood maximization in operations, while the complexity of
non-Markovian methods is in . Numerical experiments on simulated
data, as well as real-world data, show that our method enjoys improved
prediction performance when compared to state-of-the art methods like MMEL and
exponential kernels
Multivariate Hawkes Processes for Large-scale Inference
In this paper, we present a framework for fitting multivariate Hawkes
processes for large-scale problems both in the number of events in the observed
history and the number of event types (i.e. dimensions). The proposed
Low-Rank Hawkes Process (LRHP) framework introduces a low-rank approximation of
the kernel matrix that allows to perform the nonparametric learning of the
triggering kernels using at most operations, where is the
rank of the approximation (). This comes as a major improvement to
the existing state-of-the-art inference algorithms that are in .
Furthermore, the low-rank approximation allows LRHP to learn representative
patterns of interaction between event types, which may be valuable for the
analysis of such complex processes in real world datasets. The efficiency and
scalability of our approach is illustrated with numerical experiments on
simulated as well as real datasets.Comment: 16 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
Bayesian Nonparametrics to Model Content, User, and Latent Structure in Hawkes Processes
Communication in social networks tends to exhibit complex dynamics both in terms of the users involved and the contents exchanged. For example, email exchanges or activities on social media may exhibit reinforcing dynamics, where earlier events trigger follow-up activity through multiple structured latent factors. Such dynamics have been previously represented using models of reinforcement and reciprocation, a canonical example being the Hawkes process (HP). However, previous HP models do not fully capture the rich dynamics of real-world activity. For example, reciprocation may be impacted by the significance and receptivity of the content being communicated, and modeling the content accurately at the individual level may require identification and exploitation of the latent hierarchical structure present among users. Additionally, real-world activity may be driven by multiple latent triggering factors shared by past and future events, with the latent features themselves exhibiting temporal dependency structures. These important characteristics have been largely ignored in previous work. In this dissertation, we address these limitations via three novel Bayesian nonparametric Hawkes process models, where the synergy between Bayesian nonparametric models and Hawkes processes captures the structural and the temporal dynamics of communication in a unified framework. Empirical results demonstrate that our models outperform competing state-of-the-art methods, by more accurately capturing the rich dynamics of the interactions and influences among users and events, and by improving predictions about future event times, user clusters, and latent features in various types of communication activities
FaDIn: Fast Discretized Inference for Hawkes Processes with General Parametric Kernels
Temporal point processes (TPP) are a natural tool for modeling event-based
data. Among all TPP models, Hawkes processes have proven to be the most widely
used, mainly due to their simplicity and computational ease when considering
exponential or non-parametric kernels. Although non-parametric kernels are an
option, such models require large datasets. While exponential kernels are more
data efficient and relevant for certain applications where events immediately
trigger more events, they are ill-suited for applications where latencies need
to be estimated, such as in neuroscience. This work aims to offer an efficient
solution to TPP inference using general parametric kernels with finite support.
The developed solution consists of a fast L2 gradient-based solver leveraging a
discretized version of the events. After supporting the use of discretization
theoretically, the statistical and computational efficiency of the novel
approach is demonstrated through various numerical experiments. Finally, the
effectiveness of the method is evaluated by modeling the occurrence of
stimuli-induced patterns from brain signals recorded with
magnetoencephalography (MEG). Given the use of general parametric kernels,
results show that the proposed approach leads to a more plausible estimation of
pattern latency compared to the state-of-the-art
Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity
The multivariate Hawkes process is a past-dependent point process used to
model the relationship of event occurrences between different
phenomena.Although the Hawkes process was originally introduced to describe
excitation effects, which means that one event increases the chances of another
occurring, there has been a growing interest in modelling the opposite effect,
known as inhibition.In this paper, we focus on how to infer the parameters of a
multidimensional exponential Hawkes process with both excitation and inhibition
effects. Our first result is to prove the identifiability of this model under a
few sufficient assumptions. Then we propose a maximum likelihood approach to
estimate the interaction functions, which is, to the best of our knowledge, the
first exact inference procedure in the frequentist framework.Our method
includes a variable selection step in order to recover the support of
interactions and therefore to infer the connectivity graph.A benefit of our
method is to provide an explicit computation of the log-likelihood, which
enables in addition to perform a goodness-of-fit test for assessing the quality
of estimations.We compare our method to standard approaches, which were
developed in the linear framework and are not specifically designed for
handling inhibiting effects.We show that the proposed estimator performs better
on synthetic data than alternative approaches. We also illustrate the
application of our procedure to a neuronal activity dataset, which highlights
the presence of both exciting and inhibiting effects between neurons.Comment: Statistics and Computing, 202