3,530 research outputs found
The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument
We present new algorithms that efficiently approximate the hypergeometric
function of a matrix argument through its expansion as a series of Jack
functions. Our algorithms exploit the combinatorial properties of the Jack
function, and have complexity that is only linear in the size of the matrix.Comment: 14 pages, 3 figure
Evaluating parametric holonomic sequences using rectangular splitting
We adapt the rectangular splitting technique of Paterson and Stockmeyer to
the problem of evaluating terms in holonomic sequences that depend on a
parameter. This approach allows computing the -th term in a recurrent
sequence of suitable type using "expensive" operations at the cost
of an increased number of "cheap" operations.
Rectangular splitting has little overhead and can perform better than either
naive evaluation or asymptotically faster algorithms for ranges of
encountered in applications. As an example, fast numerical evaluation of the
gamma function is investigated. Our work generalizes two previous algorithms of
Smith.Comment: 8 pages, 2 figure
Recursive Estimation of Orientation Based on the Bingham Distribution
Directional estimation is a common problem in many tracking applications.
Traditional filters such as the Kalman filter perform poorly because they fail
to take the periodic nature of the problem into account. We present a recursive
filter for directional data based on the Bingham distribution in two
dimensions. The proposed filter can be applied to circular filtering problems
with 180 degree symmetry, i.e., rotations by 180 degrees cannot be
distinguished. It is easily implemented using standard numerical techniques and
suitable for real-time applications. The presented approach is extensible to
quaternions, which allow tracking arbitrary three-dimensional orientations. We
evaluate our filter in a challenging scenario and compare it to a traditional
Kalman filtering approach
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Toward an efficiently computable formula for the output statistics of MIMO block-fading channels
The information that can be conveyed through a wireless channel, with multiple-antenna equipped transmitter and receiver, crucially depends on the channel behavior as well as on the input structure. In this paper, we present very recent analytical results, concerning the probability density function (pdf) of the output of a single-user, multiple-antenna communication. The analysis is carried out under the assumption of an optimized input structure, and assuming Gaussian noise and block-fading. A further simplification of the output pdf expression presented in our last paper is derived, without the need for resorting to involved integration rules over unitary matrices. With respect to the former result, presented at the main track of this conference, the newly derived formula has the appealing feature of being numerically implementable with open access Matlab codes developed at MIT for the evaluation of zonal polynomial
The affinely invariant distance correlation
Sz\'{e}kely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769-2794) and
Sz\'{e}kely and Rizzo (Ann. Appl. Statist. 3 (2009) 1236-1265), in two seminal
papers, introduced the powerful concept of distance correlation as a measure of
dependence between sets of random variables. We study in this paper an affinely
invariant version of the distance correlation and an empirical version of that
distance correlation, and we establish the consistency of the empirical
quantity. In the case of subvectors of a multivariate normally distributed
random vector, we provide exact expressions for the affinely invariant distance
correlation in both finite-dimensional and asymptotic settings, and in the
finite-dimensional case we find that the affinely invariant distance
correlation is a function of the canonical correlation coefficients. To
illustrate our results, we consider time series of wind vectors at the
Stateline wind energy center in Oregon and Washington, and we derive the
empirical auto and cross distance correlation functions between wind vectors at
distinct meteorological stations.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ558 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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