225 research outputs found
Causal conditioning and instantaneous coupling in causality graphs
The paper investigates the link between Granger causality graphs recently
formalized by Eichler and directed information theory developed by Massey and
Kramer. We particularly insist on the implication of two notions of causality
that may occur in physical systems. It is well accepted that dynamical
causality is assessed by the conditional transfer entropy, a measure appearing
naturally as a part of directed information. Surprisingly the notion of
instantaneous causality is often overlooked, even if it was clearly understood
in early works. In the bivariate case, instantaneous coupling is measured
adequately by the instantaneous information exchange, a measure that
supplements the transfer entropy in the decomposition of directed information.
In this paper, the focus is put on the multivariate case and conditional graph
modeling issues. In this framework, we show that the decomposition of directed
information into the sum of transfer entropy and information exchange does not
hold anymore. Nevertheless, the discussion allows to put forward the two
measures as pillars for the inference of causality graphs. We illustrate this
on two synthetic examples which allow us to discuss not only the theoretical
concepts, but also the practical estimation issues.Comment: submitte
A Primer on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces are elucidated without assuming prior
familiarity with Hilbert spaces. Compared with extant pedagogic material,
greater care is placed on motivating the definition of reproducing kernel
Hilbert spaces and explaining when and why these spaces are efficacious. The
novel viewpoint is that reproducing kernel Hilbert space theory studies
extrinsic geometry, associating with each geometric configuration a canonical
overdetermined coordinate system. This coordinate system varies continuously
with changing geometric configurations, making it well-suited for studying
problems whose solutions also vary continuously with changing geometry. This
primer can also serve as an introduction to infinite-dimensional linear algebra
because reproducing kernel Hilbert spaces have more properties in common with
Euclidean spaces than do more general Hilbert spaces.Comment: Revised version submitted to Foundations and Trends in Signal
Processin
Causalité de Granger pour des signaux à valeurs fonctionnelles
National audienceGranger causality is an answer to knowing whether a signal influences another signal or not. Its principle relies on prediction theory: a first signal causes a second one if the first helps in the prediction of the second. Granger causality was developed for economy applications but is now widely used across many fields, however in stationary situations. Possible extension have already been proposed to tackle nonstationarity, but these are ratherad-hoc. A general theory for Granger causality in nonstationary contexts is still lacking. Motivated by cyclostationary signals, we propose here to model signals as discrete time signals taking values in infinite dimensional Hilbert function spaces. We then extend Granger causality in this particular context, providing strong and weak definitions, and giving some hints to practically apply the theor
Stochastic discrete scale invariance: Renormalization group operators and Iterated Function Systems
International audienceWe revisit here the notion of discrete scale invariance. Initially defined for signal indexed by the positive reals, we present a generalized version of discrete scale invariant signals relying on a renormalization group approach. In this view, the signals are seen as fixed point of a renormalization operator acting on a space of signal. We recall how to show that these fixed point present discrete scale invariance. As an illustration we use the random iterated function system as generators of random processes of the interval that are dicretely scale invariant
Wavelet analysis of the multivariate fractional Brownian motion
The work developed in the paper concerns the multivariate fractional Brownian
motion (mfBm) viewed through the lens of the wavelet transform. After recalling
some basic properties on the mfBm, we calculate the correlation structure of
its wavelet transform. We particularly study the asymptotic behavior of the
correlation, showing that if the analyzing wavelet has a sufficient number of
null first order moments, the decomposition eliminates any possible long-range
(inter)dependence. The cross-spectral density is also considered in a second
part. Its existence is proved and its evaluation is performed using a von
Bahr-Essen like representation of the function \sign(t) |t|^\alpha. The
behavior of the cross-spectral density of the wavelet field at the zero
frequency is also developed and confirms the results provided by the asymptotic
analysis of the correlation
Projections of determinantal point processes
Let be a space filling-design of
points defined in . In computer experiments, an important property
seeked for is a nice coverage of . This property could
be desirable as well as for any projection of onto
for . Thus we expect that , which represents the design
with coordinates associated to any index set , remains
regular in where is the cardinality of . This paper
examines the conservation of nice coverage by projection using spatial point
processes, and more specifically using the class of determinantal point
processes. We provide necessary conditions on the kernel defining these
processes, ensuring that the projected point process is
repulsive, in the sense that its pair correlation function is uniformly bounded
by 1, for all . We present a few examples, compare
them using a new normalized version of Ripley's function. Finally, we
illustrate the interest of this research for Monte-Carlo integration
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