808 research outputs found
Data-driven inference on optimal input-output properties of polynomial systems with focus on nonlinearity measures
In the context of dynamical systems, nonlinearity measures quantify the
strength of nonlinearity by means of the distance of their input-output
behaviour to a set of linear input-output mappings. In this paper, we establish
a framework to determine nonlinearity measures and other optimal input-output
properties for nonlinear polynomial systems without explicitly identifying a
model but from a finite number of input-state measurements which are subject to
noise. To this end, we deduce from data for the unidentified ground-truth
system three possible set-membership representations, compare their accuracy,
and prove that they are asymptotically consistent with respect to the amount of
samples. Moreover, we leverage these representations to compute guaranteed
upper bounds on nonlinearity measures and the corresponding optimal linear
approximation model via semi-definite programming. Furthermore, we extend the
established framework to determine optimal input-output properties described by
time domain hard integral quadratic constraints
Detecting and quantifying causal associations in large nonlinear time series datasets
Identifying causal relationships and quantifying their strength from observational time series data are key problems in disciplines dealing with complex dynamical systems such as the Earth system or the human body. Data-driven causal inference in such systems is challenging since datasets are often high dimensional and nonlinear with limited sample sizes. Here, we introduce a novel method that flexibly combines linear or nonlinear conditional independence tests with a causal discovery algorithm to estimate causal networks from large-scale time series datasets. We validate the method on time series of well-understood physical mechanisms in the climate system and the human heart and using large-scale synthetic datasets mimicking the typical properties of real-world data. The experiments demonstrate that our method outperforms state-of-the-art techniques in detection power, which opens up entirely new possibilities to discover and quantify causal networks from time series across a range of research fields
Robust Network Topology Inference and Processing of Graph Signals
The abundance of large and heterogeneous systems is rendering contemporary
data more pervasive, intricate, and with a non-regular structure. With
classical techniques facing troubles to deal with the irregular (non-Euclidean)
domain where the signals are defined, a popular approach at the heart of graph
signal processing (GSP) is to: (i) represent the underlying support via a graph
and (ii) exploit the topology of this graph to process the signals at hand. In
addition to the irregular structure of the signals, another critical limitation
is that the observed data is prone to the presence of perturbations, which, in
the context of GSP, may affect not only the observed signals but also the
topology of the supporting graph. Ignoring the presence of perturbations, along
with the couplings between the errors in the signal and the errors in their
support, can drastically hinder estimation performance. While many GSP works
have looked at the presence of perturbations in the signals, much fewer have
looked at the presence of perturbations in the graph, and almost none at their
joint effect. While this is not surprising (GSP is a relatively new field), we
expect this to change in the upcoming years. Motivated by the previous
discussion, the goal of this thesis is to advance toward a robust GSP paradigm
where the algorithms are carefully designed to incorporate the influence of
perturbations in the graph signals, the graph support, and both. To do so, we
consider different types of perturbations, evaluate their disruptive impact on
fundamental GSP tasks, and design robust algorithms to address them.Comment: Dissertatio
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