526 research outputs found
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
On photon statistics parametrized by a non-central Wishart random matrix
In order to tackle parameter estimation of photocounting distributions,
polykays of acting intensities are proposed as a new tool for computing photon
statistics. As unbiased estimators of cumulants, polykays are computationally
feasible thanks to a symbolic method recently developed in dealing with
sequences of moments. This method includes the so-called method of moments for
random matrices and results to be particularly suited to deal with convolutions
or random summations of random vectors. The overall photocounting effect on a
deterministic number of pixels is introduced. A random number of pixels is also
considered. The role played by spectral statistics of random matrices is
highlighted in approximating the overall photocounting distribution when acting
intensities are modeled by a non-central Wishart random matrix. Generalized
complete Bell polynomials are used in order to compute joint moments and joint
cumulants of multivariate photocounters. Multivariate polykays can be
successfully employed in order to approximate the multivariate Mendel-Poisson
transform. Open problems are addressed at the end of the paper.Comment: 18 pages, in press in Journal of Statistical Planning and Inference,
201
Rethinking LDA: moment matching for discrete ICA
We consider moment matching techniques for estimation in Latent Dirichlet
Allocation (LDA). By drawing explicit links between LDA and discrete versions
of independent component analysis (ICA), we first derive a new set of
cumulant-based tensors, with an improved sample complexity. Moreover, we reuse
standard ICA techniques such as joint diagonalization of tensors to improve
over existing methods based on the tensor power method. In an extensive set of
experiments on both synthetic and real datasets, we show that our new
combination of tensors and orthogonal joint diagonalization techniques
outperforms existing moment matching methods.Comment: 30 pages; added plate diagrams and clarifications, changed style,
corrected typos, updated figures. in Proceedings of the 29-th Conference on
Neural Information Processing Systems (NIPS), 201
On a symbolic representation of non-central Wishart random matrices with applications
By using a symbolic method, known in the literature as the classical umbral
calculus, the trace of a non-central Wishart random matrix is represented as
the convolution of the trace of its central component and of a formal variable
involving traces of its non-centrality matrix. Thanks to this representation,
the moments of this random matrix are proved to be a Sheffer polynomial
sequence, allowing us to recover several properties. The multivariate symbolic
method generalizes the employment of Sheffer representation and a closed form
formula for computing joint moments and cumulants (also normalized) is given.
By using this closed form formula and a combinatorial device, known in the
literature as necklace, an efficient algorithm for their computations is set
up. Applications are given to the computation of permanents as well as to the
characterization of inherited estimators of cumulants, which turn useful in
dealing with minors of non-central Wishart random matrices. An asymptotic
approximation of generalized moments involving free probability is proposed.Comment: Journal of Multivariate Analysis (2014
Natural statistics for spectral samples
Spectral sampling is associated with the group of unitary transformations
acting on matrices in much the same way that simple random sampling is
associated with the symmetric group acting on vectors. This parallel extends to
symmetric functions, k-statistics and polykays. We construct spectral
k-statistics as unbiased estimators of cumulants of trace powers of a suitable
random matrix. Moreover we define normalized spectral polykays in such a way
that when the sampling is from an infinite population they return products of
free cumulants.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1107 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The large-scale correlations of multi-cell densities and profiles, implications for cosmic variance estimates
In order to quantify the error budget in the measured probability
distribution functions of cell densities, the two-point statistics of cosmic
densities in concentric spheres is investigated. Bias functions are introduced
as the ratio of their two-point correlation function to the two-point
correlation of the underlying dark matter distribution. They describe how cell
densities are spatially correlated. They are computed here via the so-called
large deviation principle in the quasi-linear regime. Their large-separation
limit is presented and successfully compared to simulations for density and
density slopes: this regime is shown to be rapidly reached allowing to get
sub-percent precision for a wide range of densities and variances. The
corresponding asymptotic limit provides an estimate of the cosmic variance of
standard concentric cell statistics applied to finite surveys. More generally,
no assumption on the separation is required for some specific moments of the
two-point statistics, for instance when predicting the generating function of
cumulants containing any powers of concentric densities in one location and one
power of density at some arbitrary distance from the rest. This exact "one
external leg" cumulant generating function is used in particular to probe the
rate of convergence of the large-separation approximation.Comment: 17 pages, 10 figures, replaced to match the MNRAS accepted versio
Integral correlation measures for multiparticle physics
We report on a considerable improvement in the technique of measuring
multiparticle correlations via integrals over correlation functions. A
modification of measures used in the characterization of chaotic dynamical
sytems permits fast and flexible calculation of factorial moments and cumulants
as well as their differential versions. Higher order correlation integral
measurements even of large multiplicity events such as encountered in heavy ion
collisons are now feasible. The change from ``ordinary'' to ``factorial''
powers may have important consequences in other fields such as the study of
galaxy correlations and Bose-Einstein interferometry.Comment: 23 pages, 6 tar-compressed uuencoded PostScript figures appended,
preprint TPR-92-4
A unifying framework for -statistics, polykays and their multivariate generalizations
Through the classical umbral calculus, we provide a unifying syntax for
single and multivariate -statistics, polykays and multivariate polykays.
From a combinatorial point of view, we revisit the theory as exposed by
Stuart and Ord, taking into account the Doubilet approach to symmetric
functions. Moreover, by using exponential polynomials rather than set
partitions, we provide a new formula for -statistics that results in a very
fast algorithm to generate such estimators.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6163 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
On the computation of classical, boolean and free cumulants
This paper introduces a simple and computationally efficient algorithm for
conversion formulae between moments and cumulants. The algorithm provides just
one formula for classical, boolean and free cumulants. This is realized by
using a suitable polynomial representation of Abel polynomials. The algorithm
relies on the classical umbral calculus, a symbolic language introduced by Rota
and Taylor in 1994, that is particularly suited to be implemented by using
software for symbolic computations. Here we give a MAPLE procedure. Comparisons
with existing procedures, especially for conversions between moments and free
cumulants, as well as examples of applications to some well-known distributions
(classical and free) end the paper.Comment: 14 pages. in press, Applied Mathematics and Computatio
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