Monte Carlo exact goodness‐of‐fit tests for nonhomogeneous Poisson processes

Nonhomogeneous Poisson processes (NHPPs) are often used to model failure data from repairable systems, and there is thus a need to check model fit for such models. We study the problem of obtaining exact goodness‐of‐fit tests for parametric NHPPs. The idea is to use conditional tests given a sufficient statistic under the null hypothesis model. The tests are performed by simulating conditional samples given the sufficient statistic. Algorithms are presented for testing goodness‐of‐fit for the power law and the log‐linear law NHPP models. It is noted that while exact algorithms for the power law case are well known in the literature, the availability of such algorithms for the log‐linear case seems to be less known. A data example, as well as simulations, are considered. Copyright © 2010 John Wiley & Sons, Ltd.     

Random rates of growth and return: introducing the expo‐normal distribution

Expected values and standard deviations of the geometric means of independent positive random variables are useful indicators of the long‐term profitability of an investment, or survival of a biological population. Often these quantities cannot be evaluated in a closed form, or even if they can, there may be a choice between several probability models for the ‘annual’ growth factors. This paper formulates approximations for geometric means and standard deviations. It evaluates their performance and compares the best of them with the exact values for selected probability models of the annual factors. Among these is a new model for annual log‐returns, called the expo‐normal law. This is the law of log X, where X has a normal law conditioned on X>0. Its properties are developed in some detail. It is found that for the ranges of annual means and standard deviations typically encountered in financial applications, the longer horizon values depend little on the choice of probability model, and that, where possible, exact evaluation is computationally simpler than using approximations. Copyright © 2002 John Wiley & Sons, Ltd.     

Scale‐free power‐laws as interaction between progress and diffusion

While scale‐free power‐laws are frequently found in social and technological systems, their authenticity, origin, and gained insights are often questioned, and rightfully so. The article presents a newly found rank‐frequency power‐law that aligns the top‐500 supercomputers according to their performance. Pursuing a cautious approach in a systematic way, we check for authenticity, evaluate several potential generative mechanisms, and ask the “so what” question. We evaluate and finally reject the applicability of well‐known potential generative mechanisms such as preferential attachment, self‐organized criticality, optimization, and random observation. Instead, the microdata suggest that an inverse relationship between exponential technological progress and exponential technology diffusion through social networks results in the identified fat‐tail distribution. This newly identified generative mechanism suggests that the supply and demand of technology (“technology push” and “demand pull”) align in exponential synchronicity, providing predictive insights into the evolution of highly uncertain technology markets. © 2013 Wiley Periodicals, Inc. Complexity 19: 56–65, 2014     

Analyses of baby name popularity distribution in U.S. for the last 131 years

We examine the complete dataset of baby name popularity collected by U.S. Social Security Administration for the last 131 years (1880–2010). The ranked baby name popularity can be fitted empirically by a piecewise function consisting of Beta function for the high‐ranking names and power‐law function for low‐ranking names, but not power‐law (Zipf's law) or Beta function by itself. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

Complex equiangular tight frames and erasures

In this paper we demonstrate that there are distinct differences between real and complex equiangular tight frames (ETFs) with regards to erasures. For example, we prove that there exist arbitrarily large non-trivial complex equiangular tight frames which are optimal against three erasures, and that such frames come from a unique class of complex ETFs. In addition, we extend certain results in Bodmann and Paulsen (2005) [2] to complex vector spaces as well as show that other results regarding real ETFs are not valid for complex ETFs.     

A note on constant proportion trading strategies

We consider constant proportion (CP) trading strategies when there are multiple underlying securities and use a recently derived expression for the terminal wealth of a CP strategy to address two issues. First, we characterize the performance of a CP strategy relative to the performance of the corresponding buy-and-hold strategy. We then explain the performance of leveraged ETFs which have been criticized for not performing as expected, particularly during the financial crisis of 2008.     

Gliding motion of bacteria on power‐law slime

The present investigation deals with an undulating surface model for the motility of bacteria gliding on a layer of non‐Newtonian slime. The slime being the viscoelastic material is considered as a power‐law fluid. A hydrodynamical model of motility involving an undulating cell surface which transmits stresses through a layer of exuded slime to the substratum is examined. The non‐linear differential equation resulting from the balance of momentum and mass is solved numerically by a finite difference method with an iteration technique. The manner in which the various exponent values of the power‐law flow affect the structure of the boundary layer is delineated. A comparison is made of the power‐law fluid with the Newtonian fluid. For the power‐law fluid with respect to different power‐law exponent values, shear‐thinning and shear‐thickening effects can be observed, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

This work deals with log‐symmetric regression models, which are particularly useful when the response variable is continuous, strictly positive, and following an asymmetric distribution, with the possibility of modeling atypical observations by means of robust estimation. In these regression models, the distribution of the random errors is a member of the log‐symmetric family, which is composed by the log‐contaminated‐normal, log‐hyperbolic, log‐normal, log‐power‐exponential, log‐slash and log‐Student‐t distributions, among others. One way to select the best family member in log‐symmetric regression models is using information criteria. In this paper, we formulate log‐symmetric regression models and conduct a Monte Carlo simulation study to investigate the accuracy of popular information criteria, as Akaike, Bayesian, and Hannan‐Quinn, and their respective corrected versions to choose adequate log‐symmetric regressions models. As a business application, a movie data set assembled by authors is analyzed to compare and obtain the best possible log‐symmetric regression model for box offices. The results provide relevant information for model selection criteria in log‐symmetric regressions and for the movie industry. Economic implications of our study are discussed after the numerical illustrations.     

Global regularity of the elastic fields of a power‐law model on Lipschitz domains

In this paper, we study the global regularity of the displacement and stress fields of a nonlinear elastic model of power‐law type. It is assumed that the underlying domains are Lipschitz domains which satisfy an additional geometric condition near those points, where the type of the boundary conditions changes. The proof of the global regularity result relies on a difference quotient technique. Finally, a global regularity result for the stress fields of the elastic, perfect plastic Hencky model is derived. This model appears as a limit model of the power‐law model. Copyright © 2006 John Wiley & Sons, Ltd.     

Stochastic kronecker graphs

A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large‐scale real‐world networks such as the web. This model simultaneously captures several well‐known properties of real‐world networks; in particular, it gives rise to a heavy‐tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real‐world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011     

Equiangular tight frames with centroidal symmetry

An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others.     

A finite element method for heat transfer of power‐law flow in channels with a transverse magnetic field

Heat transfer of a power‐law non‐Newtonian incompressible fluid in channels with porous walls has not been carefully studied using a proper numerical method despite a few constructions of approximate analytic solutions through the similarity transformation and perturbation method for Newtonian fluids (i.e. power‐law index being one). In this paper, we propose a finite element method for the thermal incompressible flow equations. The incompressible condition is treated by a penalty formulation. Numerical solutions are validated by comparing them with an approximate analytic solution of the Navier–Stokes equation in the Newtonian fluid case. Then, the method is used to simulate the heat transfer of various power‐law fluids. Additionally, unlike previous studies, we allow the thermal diffusivity to be a function of temperature gradient. The effect of different values of the parameters on the temperature and velocity is also discussed in this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Construction of k-Angle Tight Frames

Frames have become standard tools in signal processing due to their robustness against transmission errors and their resilience to noise. Equiangular tight frames (ETFs) are particularly useful and have been shown to be optimal for transmission under a certain number of erasures. Unfortunately, ETFs do not exist in many cases and are hard to construct when they do exist. However, it is known that an ETF of d + 1 vectors in a d dimensional space always exists. This article gives an explicit construction of ETFs of d + 1 vectors in a d dimensional space. This construction works for both real and complex cases and is simpler than existing methods. The absence of ETFs of arbitrary sizes in a given space leads to generalizations of ETFs. One way to do this to consider tight frames where the set of (acute) angles between pairs of vectors has k distinct values. This article presents a construction of tight frames such that for a given value of k, the angles between pairs of vectors take at most k distinct values. These tight frames can be related to regular graphs and association schemes.     

基于信用利差与Logistic回归的公司违约概率测算模型与实证研究

用Logistic模型计算公司违约概率在实际应用中存在两个问题:一是在缺乏公司违约记录数据库或违约记录数据库不典型的情况下,无法应用该模型或模型计算结果不准确;二是现有Logistic违约概率模型忽视了不同行业财务指标分布特征的差异性,导致公司违约概率计算结果的准确性降低。针对问题一,本文通过公司债券信用利差计算市场隐含的公司违约概率,在Logistic变换的基础上进一步确定Logistic线性回归的参数,使得公司违约概率的计算结果符合债券市场的实际状况。针对问题二,通过不同行业关键财务指标的单因子方差分析,证实了行业间财务指标的分布特征具有显著性差异,通过拟合优度证实了区分行业建立Logistic违约概率模型可显著提高违约概率测算的准确性。本文Logistic违约概率模型的构建过程如下:通过初选财务指标的相关性分析,删除反映信息重复的财务指标;通过Logistic回归中财务指标系数的显著性检验,删除对违约概率解释能力弱的财务指标;以Logistic回归的拟合优度为标准,选取各样本行业Logistic违约概率模型的关键财务指标,建立了机械设备等5个样本行业的Logistic违约概率模型,为样本内行业公司违约概率的准确测算提供模型与方法。本文的创新与特色:一是在无套利条件下,通过公司债券信用利差计算市场隐含的公司违约概率,并对其进行Logistic变换,作为Logistic线性回归的被解释变量,解决了在缺乏公司违约记录数据情况下Logistic违约概率模型的参数估计问题;二是通过单因子方差分析方法,证实了行业间财务指标的分布特征具有显著性差异,说明应区分行业建立Logistic违约概率模型;三是通过财务指标间的相关分析删除反映信息重复的财务指标,通过财务指标系数的显著性检验删除对公司违约概率解释能力弱的财务指标,保证了Logistic违约概率模型中关键财务指标选取的合理性;四是实证研究结果表明,不同行业的Logistic违约概率模型的关键财务指标不同,同一财务指标的参数也存在显著差异。实证研究结果还表明,区分行业建立Logistic违约概率模型与不区分行业相比,前者可将拟合优度及调整后的拟合优度提高近1倍。本文研究结果对于提高公司违约概率测算的准确性具有重要参考意义,对于商业银行贷款定价、公司债券发行定价、银行信用风险管理具有重要参考意义。     

The simulation of offshore turbulent dispersion using seeded eddies

A way of representing turbulence in a two-dimensional situation is introduced appropriate to depth-independent offshore fluid mechanics. The turbulence is simulated by a collection of eddies, each of which has an analytically simple form but whose size, strength and position is governed by stochastically assigned variables. The problem addressed here is how contaminant is dispersed in such an eddy field. A number of experiments are performed whereby the eddies are seeded with marked particles that move with the fluid. The variance of these particles is monitored as time varies, and the results are compared with an assumed power law distribution. Although not a perfect fit, the results are in general accord with a power law with index between 1.5 and 2.5, which is in agreement with the observed power law of 2.34 due to Okubo, and a marked improvement on random walk models which give a variance directly proportional to time. Some further applications of this technique are discussed, namely the simulation of turbulent boundary layers and the simulation of the cascade of energy up turbulent length scales.     

Some properties of the bivariate lognormal distribution for reliability applications

In this paper, we study the bivariate lognormal distribution from a reliability point of view. The conditional distribution of X given Y > y is found to be log‐skew normal. The monotonicity of the hazard rates of the univariate as well as the conditional distributions is discussed. Clayton's association measure is obtained in terms of the hazard gradient, and its value in the case of our model is derived. The probability distributions, in the case of series and parallel systems, are derived, and the monotonicity of their failure rates is discussed. Three real applications of the bivariate lognormal distribution are provided, two from financial economics and one from reliability. Copyright © 2012 John Wiley & Sons, Ltd.     

Analysis of the nonlinear behavior of ascaris trypsin inhibitor from NMR data

The characteristics of attractor control of the conformational changes of the trypsin inhibitor induced by pH variation have been detected. The autocorrelation functions for the changes in the root-mean-square fluctuations of the atoms caused by the pH variation show the damping expected for the control of an attractor, the Poincaré plot presents a region that is preferentially occupied and the first Lyapunov coefficient is positive, λ=0.094±0.001. The power spectrum is broad and the linear fit of power spectrum represented in a log–log scale shows broadband profile with 1/f-like distributions that are typical for chaotic processes. The saturation of the integral correlation coefficient with increasing number of coordinates in the phase space indicates the presence of an attractor with the dimension 2.24±0.15.     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015