scholarly journals THE SPECTRUM OF RANDOM INNER-PRODUCT KERNEL MATRICES

2013 ◽  
Vol 02 (04) ◽  
pp. 1350010 ◽  
Author(s):  
XIUYUAN CHENG ◽  
AMIT SINGER
Keyword(s):  
Random Matrices ◽  
Kernel Function ◽  
Weak Limit ◽  
Kernel Functions ◽  
Inner Product ◽  
Cubic Equation ◽  
New Family ◽  
Sample Covariance ◽  
Special Cases ◽  
Product Kernel

We consider n × n matrices whose (i, j)th entry is [Formula: see text], where X1, …, Xn are i.i.d. standard Gaussian in ℝp, and f is a real-valued function. The weak limit of the eigenvalue distribution of these random matrices is studied at the limit when p, n → ∞ and p/n = γ which is a constant. We show that, under certain conditions on the kernel function f, the limiting spectral density exists and is dictated by a cubic equation involving its Stieltjes transform. The parameters of this cubic equation are decided by a Hermite-like expansion of the rescaled kernel function. While the case that f is differentiable at the origin has been previously resolved by El Karoui [The spectrum of kernel random matrices, Ann. Statist.38 (2010) 1–50], our result is applicable to non-smooth f, such as the Sign function and the hard thresholding operator of sample covariance matrices. For this larger class of kernel functions, we obtain a new family of limiting densities, which includes the Marčenko–Pastur (M.P.) distribution and Wigner's semi-circle distribution as special cases.

scholarly journals THE SPECTRUM OF RANDOM KERNEL MATRICES: UNIVERSALITY RESULTS FOR ROUGH AND VARYING KERNELS

2013 ◽  
Vol 02 (03) ◽  
pp. 1350005 ◽  
Author(s):  
YEN DO ◽  
VAN VU
Keyword(s):  
Large Class ◽  
Random Matrices ◽  
Spectral Distribution ◽  
Inner Product ◽  
Limiting Spectral Distribution ◽  
Product Kernel ◽  
Gaussian Vectors ◽  
Spectral Distributions ◽  
As If

We consider random matrices whose entries are [Formula: see text] or f(‖Xi – Xj‖2) for iid vectors Xi ∈ ℝp with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui [The spectrum of Kernel random matrices, Ann. Statist.38(1) (2010) 1–50, MR 2589315 (2011a.62187)] showed that the spectral distributions of these matrices behave as if f is linear in the Marčhenko–Pastur limit. When Xi's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng–Singer [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]]. Two results are shown in this paper: first, it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and second, it is shown that the Gaussian assumptions in Cheng–Singer's result can be removed, answering a question posed in [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]] about the universality of the limiting spectral distribution.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

scholarly journals Survival and Reliability Analysis with an Epsilon-Positive Family of Distributions with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 908
Author(s):  
Perla Celis ◽  
Rolando de la Cruz ◽  
Claudio Fuentes ◽  
Héctor W. Gómez
Keyword(s):  
Survival Data ◽  
Maximum Likelihood Estimates ◽  
New Class ◽  
New Family ◽  
Gamma Distributions ◽  
Special Cases ◽  
Log Normal ◽  
Positive Support ◽  
Positive Distributions ◽  
Family Of Distributions

We introduce a new class of distributions called the epsilon–positive family, which can be viewed as generalization of the distributions with positive support. The construction of the epsilon–positive family is motivated by the ideas behind the generation of skew distributions using symmetric kernels. This new class of distributions has as special cases the exponential, Weibull, log–normal, log–logistic and gamma distributions, and it provides an alternative for analyzing reliability and survival data. An interesting feature of the epsilon–positive family is that it can viewed as a finite scale mixture of positive distributions, facilitating the derivation and implementation of EM–type algorithms to obtain maximum likelihood estimates (MLE) with (un)censored data. We illustrate the flexibility of this family to analyze censored and uncensored data using two real examples. One of them was previously discussed in the literature; the second one consists of a new application to model recidivism data of a group of inmates released from the Chilean prisons during 2007. The results show that this new family of distributions has a better performance fitting the data than some common alternatives such as the exponential distribution.

scholarly journals A New Family of High-Order Ehrlich-Type Iterative Methods

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1855 ◽  
Author(s):  
Petko D. Proinov ◽  
Maria T. Vasileva
Keyword(s):  
Iterative Methods ◽  
Local Convergence ◽  
Semilocal Convergence ◽  
High Order ◽  
Convergence Theorems ◽  
New Family ◽  
Special Cases ◽  
Iteration Functions ◽  
Iteration Function ◽  
Local Convergence Analysis

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems.

Mixed Convolved Action Principles for Dynamics of Linear Poroelastic Continua

2015 ◽  
Author(s):  
Bradley T. Darrall ◽  
Gary F. Dargush
Keyword(s):  
Weak Form ◽  
Dynamical Problem ◽  
Inner Product ◽  
Analytical Mechanics ◽  
Convolution Operators ◽  
Dissipative Processes ◽  
Poroelastic Media ◽  
New Family ◽  
The First Variation ◽  
Mixed Approach

Although Lagrangian and Hamiltonian analytical mechanics represent perhaps the most remarkable expressions of the dynamics of a mechanical system, these approaches also come with limitations. In particular, there is inherent difficulty to represent dissipative processes and the restrictions placed on end point variations are not consistent with the definition of initial value problems. The present work on poroelastic media extends the recent formulation of a mixed convolved action to address a continuum dynamical problem with dissipation through the development of a new variational approach. The action in this proposed approach is formed by replacing the inner product in Hamilton’s principle with a time convolution. As a result, dissipative processes can be represented in a natural way and the required constraints on the variations are consistent with the actual initial and boundary conditions of the problem. The variational formulations developed here employ temporal impulses of velocity, effective stress, pore pressure and pore fluid mass flux as primary variables in this mixed approach, which also uses convolution operators and fractional calculus to achieve the desired characteristics. The resulting mixed convolved action is formulated in both the time and frequency domains to develop two new stationary principles for dynamic poroelasticity. In addition, the first variation of the action provides a temporally well-balanced weak form that leads to a new family of finite element methods in time, as well as space.

scholarly journals A Kernel-Based Approach for Biomedical Named Entity Recognition

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Rakesh Patra ◽  
Sujan Kumar Saha
Keyword(s):  
Kernel Function ◽  
Text Processing ◽  
Named Entity Recognition ◽  
Kernel Functions ◽  
Entity Recognition ◽  
Machine Learning Techniques ◽  
Support Vector ◽  
Svm Classifier ◽  
Named Entity ◽  
Tree Kernel

Support vector machine (SVM) is one of the popular machine learning techniques used in various text processing tasks including named entity recognition (NER). The performance of the SVM classifier largely depends on the appropriateness of the kernel function. In the last few years a number of task-specific kernel functions have been proposed and used in various text processing tasks, for example, string kernel, graph kernel, tree kernel and so on. So far very few efforts have been devoted to the development of NER task specific kernel. In the literature we found that the tree kernel has been used in NER task only for entity boundary detection or reannotation. The conventional tree kernel is unable to execute the complete NER task on its own. In this paper we have proposed a kernel function, motivated by the tree kernel, which is able to perform the complete NER task. To examine the effectiveness of the proposed kernel, we have applied the kernel function on the openly available JNLPBA 2004 data. Our kernel executes the complete NER task and achieves reasonable accuracy.

Inner-product theory calculations for some rational potentials in a three-dimensional system

1996 ◽  
Vol 74 (1-2) ◽  
pp. 4-9
Author(s):  
M. R. M. Witwit
Keyword(s):  
Energy Levels ◽  
Three Dimensional ◽  
Inner Product ◽  
Dimensional System ◽  
Product Technique ◽  
Special Cases ◽  
Wide Range ◽  
Perturbation Parameters ◽  
Three Dimensional System ◽  
Range Of Values

The energy levels of a three-dimensional system are calculated for the rational potentials,[Formula: see text]using the inner-product technique over a wide range of values of the perturbation parameters (λ, g) and for various eigenstates. The numerical results for some special cases agree with those of previous workers where available.

scholarly journals A COMPARISON STUDY OF DIFFERENT KERNEL FUNCTIONS FOR SVM-BASED CLASSIFICATION OF MULTI-TEMPORAL POLARIMETRY SAR DATA

2014 ◽  
Vol XL-2/W3 ◽  
pp. 281-285 ◽  
Author(s):  
B. Yekkehkhany ◽  
A. Safari ◽  
S. Homayouni ◽  
M. Hasanlou
Keyword(s):  
Kernel Function ◽  
Temporal Integration ◽  
Experimental Tests ◽  
Kernel Functions ◽  
Polynomial Kernel ◽  
Support Vector ◽  
Svm Classifier ◽  
Rbf Kernel ◽  
Multi Temporal ◽  
Dimension Space

In this paper, a framework is developed based on Support Vector Machines (SVM) for crop classification using polarimetric features extracted from multi-temporal Synthetic Aperture Radar (SAR) imageries. The multi-temporal integration of data not only improves the overall retrieval accuracy but also provides more reliable estimates with respect to single-date data. Several kernel functions are employed and compared in this study for mapping the input space to higher Hilbert dimension space. These kernel functions include linear, polynomials and Radial Based Function (RBF). <br><br> The method is applied to several UAVSAR L-band SAR images acquired over an agricultural area near Winnipeg, Manitoba, Canada. In this research, the temporal alpha features of H/A/α decomposition method are used in classification. The experimental tests show an SVM classifier with RBF kernel for three dates of data increases the Overall Accuracy (OA) to up to 3% in comparison to using linear kernel function, and up to 1% in comparison to a 3rd degree polynomial kernel function.

scholarly journals The modified beta transmuted family of distributions with applications using the exponential distribution

PLoS ONE ◽  
2021 ◽  
Vol 16 (11) ◽  
pp. e0258512
Author(s):  
Phillip Oluwatobi Awodutire ◽  
Oluwafemi Samson Balogun ◽  
Akintayo Kehinde Olapade ◽  
Ethelbert Chinaka Nduka
Keyword(s):  
Exponential Distribution ◽  
Real Life ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Life Time ◽  
Time Data ◽  
New Family ◽  
Special Cases ◽  
The Family ◽  
Family Of Distributions

In this work, a new family of distributions, which extends the Beta transmuted family, was obtained, called the Modified Beta Transmuted Family of distribution. This derived family has the Beta Family of Distribution and the Transmuted family of distribution as subfamilies. The Modified beta transmuted frechet, modified beta transmuted exponential, modified beta transmuted gompertz and modified beta transmuted lindley were obtained as special cases. The analytical expressions were studied for some statistical properties of the derived family of distribution which includes the moments, moments generating function and order statistics. The estimates of the parameters of the family were obtained using the maximum likelihood estimation method. Using the exponential distribution as a baseline for the family distribution, the resulting distribution (modified beta transmuted exponential distribution) was studied and its properties. The modified beta transmuted exponential distribution was applied to a real life time data to assess its flexibility in which the results shows a better fit when compared to some competitive models.

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