scholarly journals High-dimensional asymptotic behavior of the difference between the log-determinants of two Wishart matrices

2017 ◽  
Vol 157 ◽  
pp. 70-86 ◽  
Author(s):  
Hirokazu Yanagihara ◽  
Ryoya Oda ◽  
Yusuke Hashiyama ◽  
Yasunori Fujikoshi
Keyword(s):  
Asymptotic Behavior ◽  
High Dimensional ◽  
Wishart Matrices ◽  
The Difference

scholarly journals ON DETERMINING THE NUMBER OF SPIKES IN A HIGH-DIMENSIONAL SPIKED POPULATION MODEL

2012 ◽  
Vol 01 (01) ◽  
pp. 1150002 ◽  
Author(s):  
DAMIEN PASSEMIER ◽  
JIAN-FENG YAO
Keyword(s):  
Covariance Matrix ◽  
Population Model ◽  
Factor Model ◽  
Fundamental Problem ◽  
High Dimensional ◽  
Sample Covariance ◽  
Or Economics ◽  
Linear Mixture Model ◽  
The Difference ◽  
Scientific Fields

In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). Determining the number of spikes is a fundamental problem which appears in many scientific fields, including signal processing (linear mixture model) or economics (factor model). Several recent papers studied the asymptotic behavior of the eigenvalues of the sample covariance matrix (sample eigenvalues) when the dimension of the observations and the sample size both grow to infinity so that their ratio converges to a positive constant. Using these results, we propose a new estimator based on the difference between two consecutive sample eigenvalues.

FRACTAL METHODS APPLIED TO DESCRIBE CELLULAR AUTOMATON COMBAT MODELS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 177-184 ◽  
Author(s):  
MICHAEL K. LAUREN
Keyword(s):  
Cellular Automaton ◽  
Complexity Theory ◽  
Power Law ◽  
Chaotic Systems ◽  
Probability Distributions ◽  
High Dimensional ◽  
Compelling Evidence ◽  
Military Equipment ◽  
The Difference ◽  
Maneuver Warfare

Analysis of warfare data provides compelling evidence that intensity of conflicts obeys a power-law (fractal) dependence on frequency. There is also evidence for the existence of other power-law dependences and traits characteristic of high-dimensional chaotic systems, such as fat-tailed probability distributions and intermittency in warfare data. In this report, it is discussed how a cellular automaton model used to describe modern maneuver warfare produces casualty distributions which exhibit these properties. This points to a possible origin of the characteristics of the larger timescale data. More interesting, the techniques of fractal analysis offer a method by which to characterize these behaviors, and to quantify the difference between models based on complexity theory (such as cellular automata models), and more traditional combat models based on the physics of military equipment.

Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

scholarly journals Global Behavior of a Discrete Anticompetitive System in the Plane

2019 ◽  
Vol 2019 ◽  
pp. 1-7
Author(s):  
Lin-Xia Hu
Keyword(s):  
Asymptotic Behavior ◽  
Open Problem ◽  
Global Attractivity ◽  
Global Behavior ◽  
Initial Condition ◽  
Difference System ◽  
Rational Systems ◽  
The Difference

The main goal of this paper is to investigate the global asymptotic behavior of the difference system xn+1=γ1yn/A1+xn,  yn+1=β2xn/B2+yn,  n=0,1,2,…. with γ1,β2,A1,B2∈(0,∞) and the initial condition (x0,y0)∈[0,∞)×[0,∞). We obtain some global attractivity results of this system for different values of the parameters, which answer the open problem proposed in “Rational systems in the plane, J. Difference Equ. Appl. 15 (2009), 303-323”.

scholarly journals Statistical Complexity of Low- and High-Dimensional Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Vladimir Ryabov ◽  
Dmitry Nerukh
Keyword(s):  
Time Series ◽  
Markov Property ◽  
Basic Mechanism ◽  
High Dimensional ◽  
Type Model ◽  
Stochastic Motion ◽  
Standard Map ◽  
Deterministic Systems ◽  
Statistical Complexity ◽  
The Difference

We suggest a new method for the analysis of experimental time series that can distinguish high-dimensional dynamics from stochastic motion. It is based on the idea of statistical complexity, that is, the Shannon entropy of the so-called ϵ-machine (a Markov-type model of the observed time series). This approach has been recently demonstrated to be efficient for making a distinction between a molecular trajectory in water and noise. In this paper, we analyse the difference between chaos and noise using the Chirikov-Taylor standard map as an example in order to elucidate the basic mechanism that makes the value of complexity in deterministic systems high. In particular, we show that the value of statistical complexity is high for the case of chaos and attains zero value for the case of stochastic noise. We further study the Markov property of the data generated by the standard map to clarify the role of long-time memory in differentiating the cases of deterministic systems and stochastic motion.

Predicting the Future: Advantages of Semilocal Units

1991 ◽  
Vol 3 (4) ◽  
pp. 566-578 ◽  
Author(s):  
Eric Hartman ◽  
James D. Keeler
Keyword(s):  
Gradient Descent ◽  
High Dimensional ◽  
Activation Functions ◽  
Learning Problem ◽  
Rbf Networks ◽  
Present Evidence ◽  
Training Time ◽  
The Difference ◽  
Low Dimensional ◽  
Slow Learning

In investigating gaussian radial basis function (RBF) networks for their ability to model nonlinear time series, we have found that while RBF networks are much faster than standard sigmoid unit backpropagation for low-dimensional problems, their advantages diminish in high-dimensional input spaces. This is particularly troublesome if the input space contains irrelevant variables. We suggest that this limitation is due to the localized nature of RBFs. To gain the advantages of the highly nonlocal sigmoids and the speed advantages of RBFs, we propose a particular class of semilocal activation functions that is a natural interpolation between these two families. We present evidence that networks using these gaussian bar units avoid the slow learning problem of sigmoid unit networks, and, very importantly, are more accurate than RBF networks in the presence of irrelevant inputs. On the Mackey-Glass and Coupled Lattice Map problems, the speedup over sigmoid networks is so dramatic that the difference in training time between RBF and gaussian bar networks is minor. Gaussian bar architectures that superpose composed gaussians (gaussians-of-gaussians) to approximate the unknown function have the best performance. We postulate that an interesing behavior displayed by gaussian bar functions under gradient descent dynamics, which we call automatic connection pruning, is an important factor in the success of this representation.

scholarly journals Dimer–monomer model on the Towers of Hanoi graphs

2015 ◽  
Vol 29 (23) ◽  
pp. 1550173 ◽  
Author(s):  
Hanlin Chen ◽  
Renfang Wu ◽  
Guihua Huang ◽  
Hanyuan Deng
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bounds ◽  
Statistical Physics ◽  
Upper And Lower Bounds ◽  
Recursion Relations ◽  
Sierpiński Graphs ◽  
Towers Of Hanoi ◽  
The Difference ◽  
Significant Figures

The number of dimer–monomers (matchings) of a graph [Formula: see text] is an important graph parameter in statistical physics. Following recent research, we study the asymptotic behavior of the number of dimer–monomers [Formula: see text] on the Towers of Hanoi graphs and another variation of the Sierpiński graphs which is similar to the Towers of Hanoi graphs, and derive the recursion relations for the numbers of dimer–monomers. Upper and lower bounds for the entropy per site, defined as [Formula: see text], where [Formula: see text] is the number of vertices in a graph [Formula: see text], on these Sierpiński graphs are derived in terms of the numbers at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accuracy.

scholarly journals Limiting probability measures

2020 ◽  
Vol 12 ◽  
Author(s):  
Irfan Alam
Keyword(s):  
Asymptotic Behavior ◽  
Classical Result ◽  
Nonstandard Analysis ◽  
High Dimensional ◽  
Probability Measures ◽  
Dimensional Sphere ◽  
Spherical Means ◽  
Fixed Direction ◽  
Set Up ◽  
Gaussian Moment

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof.

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