252,373 research outputs found
A New Nonparametric Test of Cointegration Rank
No full textThis paper suggests a new nonparametric testing procedure for determining the rank of nonstationary multivariate cointegrated systems. The asymptotic properties of the procedure are determined and a Monte Carlo study is carried out.Cointegration rank, Nonparametric analysis
Local rank tests in a multivariate nonparametric relationship
No full textConsider a multivariate nonparametric model where the unknown vector of functions depends on two sets of explanatory variables. For a fixed level of one set of explanatory variables, we provide consistent statistical tests, called local rank tests, to determine whether the multivariate relationship can be explained by a smaller number of functions. We also provide estimators for the smallest number of functions, called local rank, explaining the relationship. The local rank tests and the estimators of local rank are defined in terms of the eigenvalues of a kernel-based estimator of some matrix. The asymptotics of the eigenvalues is established by using the so-called Fujikoshi expansion along with some techniques of the theory of U-statistics. We present a simulation study which examines the small sample properties of local rank tests. We also apply the local rank tests and the local rank estimators to a demand system given by a newly constructed data set. This work can be viewed as a “local” extension of the tests for a number of factors in a nonparametric relationship introduced by Stephen Donald.nonparametric relationship, local rank, local rank estimation, kernel smoothing, consistent tests, demand systems.
A solver for clustered low-rank SDPs arising from multivariate polynomial (matrix) programs
No full textIn this thesis, we give a primal-dual interior point method specialized to clustered low-rank semidefinite programs. We introduce multivariate polynomial matrix programs, and we reduce these to clustered low-rank semidefinite programs. This extends the work of Simmons-Duffin [J. High Energ. Phys. 1506, no. 174 (2015)] from univariate to multivariate polynomial matrix programs, and to more general clustered low-rank semidefinite programs. Clustered low-rank semidefinite programs are programs with low-rank constraint matrices where the positive semidefinite variables are only used within clusters of constraints. The free variables can be used in any constraint, and can be used to connect clusters. The solver uses this structure to speed up the computations in two ways. First, the low rank structure is used to reduce matrix products to products of the form uT M v, where M is a matrix and u and v are vectors, as already suggested by Löfberg and Parrilo in [43rd IEEE CDC (2004)]. Second, an additional block-diagonal structure is introduced due to the clusters. This gives the possibility to do operations such as the Cholesky decomposition block-wise. We apply this to sphere packing and spherical cap packing. For sphere packing, the speed of the solver is compared to the often used arbitrary precision solver SDPA-GMP, showing the theoretical speedup in time complexity. We give a new three-point bound for the maximum density when packing spherical caps of sizes on the unit sphere. https://github.com/nanleij/Clustered-Low-Rank-SDP-solver Repository link Github repository with the Julia code of the solverApplied Mathematics | Optimizatio
COMPARING THE EFFECTIVENESS OF RANK CORRELATION STATISTICS
No full textRank correlation is a fundamental tool to express dependence in cases in which the data are arranged in order. There are, by contrast, circumstances where the ordinal association is of a nonlinear type. In this paper we investigate the effectiveness of several measures of rank correlation. These measures have been divided into three classes: conventional rank correlations, weighted rank correlations, correlations of scores. Our findings suggest that none is systematically better than the other in all circumstances. However, a simply weighted version of the Kendall rank correlation coefficient provides plausible answers to many special situations where intercategory distances could not be considered on the same basis.Ordinal Data, Nonlinear Association, Weighted Rank Correlation
Representing Symmetric Rank Two Updates
No full textVarious quasi-Newton methods periodically add a symmetric "correction" matrix of rank at most 2 to a matrix approximating some quantity A of interest (such as the Hessian of an objective function). In this paper we examine several ways to express a symmetric rank 2 matrix [delta] as the sum of rank 1 matrices. We show that it is easy to compute rank 1 matrices [delta1] and [delta2] such that [delta] = [delta1] + [delta2] and [the norm of delta1]+ [the norm of delta2] is minimized, where ||.|| is any inner product norm. Such a representation recommends itself for use in those computer programs that maintain A explicitly, since it should reduce cancellation errors and/or improve efficiency over other representations. In the common case where [delta] is indefinite, a choice of the form [delta1] = [delta2 to the power of T] = [xy to the power of T] appears best. This case occurs for rank 2 quasi- Newton updates [delta] exactly when [delta] may be obtained by symmetrizing some rank 1 update; such popular updates as the DFP, BFGS, PSB, and Davidon's new optimally conditioned update fall into this category.
On rank estimation in semidefinite matrices
No full textThis work concerns the problem of rank estimation in semidefinite matrices, having either indefinite or semidefinite matrix estimator satisfying a typical asymptotic normality condition. Several rank tests are examined, based on either available rank tests or basic new results. A number of related issues are discussed such as the choice of matrix estimators and rank tests based on finer assumptions than those of asymptotic normality of matrix estimators. Several examples where rank estimation in semidefinite matrices is of interest are studied and serve as guide throughout the work.rank, symmetric matrix, indefinite and semidefinite estimators, eigenvalues, matrix decompositions, estimation, asymptotic normality.
Reduced-rank adaptive least bit-error-rate detection in hybrid direct-sequence time-hopping ultrawide bandwidth systems
No full textDesign of high-efficiency low-complexity detection schemes for ultrawide bandwidth (UWB) systems is highly challenging. This contribution proposes a reduced-rank adaptive multiuser detection (MUD) scheme operated in least bit-errorrate (LBER) principles for the hybrid direct-sequence timehopping UWB (DS-TH UWB) systems. The principal component analysis (PCA)-assisted rank-reduction technique is employed to obtain a detection subspace, where the reduced-rank adaptive LBER-MUD is carried out. The reduced-rank adaptive LBERMUD is free from channel estimation and does not require the knowledge about the number of resolvable multipaths as well as the knowledge about the multipaths’ strength. In this contribution, the BER performance of the hybrid DS-TH UWB systems using the proposed detection scheme is investigated, when assuming communications over UWB channels modeled by the Saleh-Valenzuela (S-V) channel model. Our studies and performance results show that, given a reasonable rank of the detection subspace, the reduced-rank adaptive LBER-MUD is capable of efficiently mitigating the multiuser interference (MUI) and inter-symbol interference (ISI), and achieving the diversity gain promised by the UWB systems
Low-Complexity Reduced-Rank Adaptive Detection in Hybrid Direct-Sequence Time-Hopping Ultrawide Bandwidth Systems
No full textAbstract — In this contribution reduced-rank adaptive minimum mean-square error multiuser detector (MMSE-MUD) is proposed and investi-gated for the hybrid direct-sequence time-hopping ultrawide bandwidth (DS-TH UWB) systems. The adaptive MMSE-MUD is operated based on the normalised least mean-square (NLMS) principles associated with using Taylor polynomial approximation (TPA)-assisted reduced-rank technique. It can be shown that the reduced-rank adaptive technique is beneficial to achieving low-complexity, high convergence speed and robust detection in hybrid DS-TH UWB systems. In this contribution bit-error-rate (BER) performance of the hybrid DS-TH UWB systems using proposed reduced-rank adaptive MMSE-MUD is investigated, when communicating over UWB channels modelled by the Saleh-Valenzuela (S-V) channel model. Our simulation results show that the TPA-assisted reduced-rank adaptive MMSE-MUD is capable of achieving a similar BER performance as that of the full-rank adaptive MMSE-MUD but with significantly lower detection complexity. I
Determination of the rank of an integration lattice
No full textThe continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations.
In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B
AN EXHAUSTIVE COEFFICIENT OF RANK CORRELATION
No full textRank association is a fundamental tool for expressing dependence in cases in which data are arranged in order. Measures of rank correlation have been accumulated in several contexts for more than a century and we were able to cite more than thirty of these coefficients, from simple ones to relatively complicated definitions invoking one or more systems of weights. However, only a few of these can actually be considered to be admissible substitutes for Pearson’s correlation. The main drawback with the vast majority of coefficients is their “resistance-tochange” which appears to be of limited value for the purposes of rank comparisons that are intrinsically robust. In this article, a new nonparametric correlation coefficient is defined that is based on the principle of maximization of a ratio of two ranks. In comparing it with existing rank correlations, it was found to have extremely high sensitivity to permutation patterns. We have illustrated the potential improvement that our index can provide in economic contexts by comparing published results with those obtained through the use of this new index. The success that we have had suggests that our index may have important applications wherever the discriminatory power of the rank correlation coefficient should be particularly strong.Ordinal data, Nonparametric agreement, Economic applications
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