Regularized classification for mixed continuous and categorical variables under across-location heteroscedasticity

A regularized classifier is proposed for a two-population classification problem of mixed continuous and categorical variables in a general location model(GLOM). The limiting overall expected error for the classifier is given. It can be used in an optimization search for the regularization parameters. For a heteroscedastic spherical dispersion across all locations, an asymptotic error is available which provides an alternative criterion for the optimization search. In addition, the asymptotic error can serve as a baseline for practical comparisons with other classifiers. Results based on a simulation and two real datasets are presented.     

The Covariance Adjusted Location Linear Discriminant Function for Classifying Data with Unequal Dispersion Matrices in Different Locations

Classification between two populations dealing with both continuous and binary variables is handled by splitting the problem into different locations. Given the location specified by the values of the binary variables, discrimination is performed using the continuous variables. The location probability model with homoscedastic across location conditional dispersion matrices is adopted for this problem. In this paper, we consider presence of continuous covariates with heterogeneous location conditional dispersion matrices. The continuous covariates have equal location specific mean in both populations. Conditional homoscedasticity fails when strong interaction between the continuous and binary variables is present. A plug-in covariance adjusted rule is constructed and its asymptotic distribution is derived. An asymptotic expansion for the overall error rate is given. The result is extended to include binary covariates.     

Nonparametric estimation of distributions with categorical and continuous data

In this paper we consider the problem of estimating an unknown joint distribution which is defined over mixed discrete and continuous variables. A nonparametric kernel approach is proposed with smoothing parameters obtained from the cross-validated minimization of the estimator's integrated squared error. We derive the rate of convergence of the cross-validated smoothing parameters to their ‘benchmark’ optimal values, and we also establish the asymptotic normality of the resulting nonparametric kernel density estimator. Monte Carlo simulations illustrate that the proposed estimator performs substantially better than the conventional nonparametric frequency estimator in a range of settings. The simulations also demonstrate that the proposed approach does not suffer from known limitations of the likelihood cross-validation method which breaks down with commonly used kernels when the continuous variables are drawn from fat-tailed distributions. An empirical application demonstrates that the proposed method can yield superior predictions relative to commonly used parametric models.     

Asymptotic expansions and Richardson extrapolation of approximate solutions for integro-differential equations by mixed finite element methods

In this paper asymptotic error expansions for mixed finite element approximations of the integro-differential equation are derived, and Richardson extrapolation is applied to improve the accuracy of the approximations by two different schemes with the help of an interpolation post-processing technique. The results of this paper provide new asymptotic expansions. As a by-product, we illustrate that all the approximations of higher accuracy can be used to form a class of a-posteriori error estimators for this mixed finite element method. Finally, a numerical example is provided to validate the theoretical results. This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical Methods for Convertible Bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), Tianjin Educational Committee, Liu Hui Center for Applied Mathematics of Nankai University and Tianjin University, and Tianjin University of Finance and Economics.     

Analytical and numerical aspects of a generalization of the complementary error function

In this paper we discuss analytical and numerical properties of the function , with α,β,Rz>0, which can be viewed as a generalization of the complementary error function, and in fact also as a generalization of the Kummer U-function. The function V_ν,μ(α, β, z) is used for certain values of the parameters as an approximate in a singular perturbation problem. We consider the relation with other special functions and give asymptotic expansions as well as recurrence relations. Several methods for its numerical evaluation and examples are given.     

A systematization of the saddle point method. Application to the Airy and Hankel functions

The standard saddle point method of asymptotic expansions of integrals requires to show the existence of the steepest descent paths of the phase function and the computation of the coefficients of the expansion from a function implicitly defined by solving an inversion problem. This means that the method is not systematic because the steepest descent paths depend on the phase function on hand and there is not a general and explicit formula for the coefficients of the expansion (like in Watson's Lemma for example). We propose a more systematic variant of the method in which the computation of the steepest descent paths is trivial and almost universal: it only depends on the location and the order of the saddle points of the phase function. Moreover, this variant of the method generates an asymptotic expansion given in terms of a generalized (and universal) asymptotic sequence that avoids the computation of the standard coefficients, giving an explicit and systematic formula for the expansion that may be easily implemented on a symbolic manipulation program. As an illustrative example, the well-known asymptotic expansion of the Airy function is rederived almost trivially using this method. New asymptotic expansions of the Hankel function Hn(z) for large n and z are given as non-trivial examples.     

A Time-Independent Approach for the Study of the Spectral Shift Function and an Application to Stark Hamiltonians

In this paper we develop a time-independent approach for the study of the spectral shift function (SSF for short). We apply this method for the perturbed Stark Hamiltonian. We obtain a weak and a Weyl-type asymptotics with optimal remainder estimate of the SSF of the operator pair (P = P₀ + V(x), P₀ = ? h²Δ +x₁), x = (x₁,…, x_n) where V(x) ∈ 𝒞^∞(?ⁿ, ?) decays sufficiently fast at infinity, and h is a small positive parameter. Near a non-trapping energy λ, we give a pointwise asymptotic expansions in powers of h of the derivative of the SSF, and we compute explicitly the two leading terms.     

The effect of alternative scoring methods on the analysis of rank order categorical data

The purpose of this article is to examine the effects of five univariate scoring techniques for rank order categorical data and the results of analyses using each of the techniques for five‐ and ten‐point bi‐polar adjective scales. The effect of scoring method and scale length is assessed for the resultant distance to multivariate normality, inter‐item reliability, discriminant analysis, least squares regression and logistic regression. For these data, the strongest effect of scoring was on distance to multivariate normality and determination of significant variables.     

The hazard rate of the power-quadratic exponential family of distributions

Among life distributions of exponential-type the power-quadratic distributions form a very versatile family since their hazard rates can have a wide variety of shapes, including the bathtub shape. It is shown in this article that this family can be partially ordered by its hazard rates whose precise expression and asymptotic behavior can also be obtained by using special mathematical functions.     

On the rate of convergence for multi-category classification based on convex losses

The multi-category classification algorithms play an important role in both theory and practice of machine learning.In this paper,we consider an approach to the multi-category classification based on minimizing a convex surrogate of the nonstandard misclassification loss.We bound the excess misclassification error by the excess convex risk.We construct an adaptive procedure to search the classifier and furthermore obtain its convergence rate to the Bayes rule.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.     

The Effect of Spatial Autocorrelation on the Error Rates of the Linear Discriminant Functions

The problem of classification of the realization of the intrinsically stationary multivariate Gaussian random field into one of two populations with different means and factorized covariance matrices is considered. Unknown means and the common covariance matrix of the feature vector components are estimated from the spatially correlated training samples assuming spatial correlations to be known. Two plugin linear discriminant functions (DF) are considered. The first linear DF uses the maximum likelihood (ML) estimators of means and the biasadjusted ML estimator of covariance. The second one uses usual sample means and biasadjusted sample covariance. The firstorder asymptotic expansions with respect to the inverses of training sample sizes of the expected error rate associated with two plugin DF's are presented. The numerical results obtained allow us to compare the performance of the suggested DF's. The numerical calculations are done for the exponential spatial correlation function.     

The number of nonseparable maps on the projective plane

In this paper, we study the rooted nonseparable maps on the sphere and the projective plane with the valency of root-face and the number of edges as parameters. Explicit expression of enumerating functions are obtained for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for such maps on the projective plane, from which asymptotic evaluations are derived. Moreover, if the number of edges is sufficiently large, then almost all nonseparable maps on the projective plane are not triangulation.     

An extended and tractable approach on the convergence problem of the mixed push-pull manpower model

In this paper, the asymptotic behavior of the time-homogeneous mixed push-pull manpower model is studied under the assumption that the desired stock vector and the recruitment policy are fixed over time. In the mixed push-pull manpower model, the internal mobility of a personnel system can be regulated by both pull and push transitions. Based on those characteristics, we express and examine the dynamics of the personnel system by formulating the mixed push-pull manpower model by means of particular transition matrices, which we demonstrate to have interesting properties. We show that under certain conditions the stock vector converges. An explicit analytical form for this limiting personnel stock vector is found.     

再生核希尔伯特空间连续线性泛函的范数及其应用北大核心

考虑了再生核希尔伯特空间连续线性泛函范数的表示,得到了用其范数平方等于该线性泛函连续两次作于再生核的简明表示.对于常见的Sobolev-Hibert空间而言,其再生核则可用截幂函数来表示,从而得到Sobolev-Hibert空间上连续线性泛函范数的简洁表示,以新视角解释和简化了文献中的现有结果.     

The Cauchy Transform of Continuous Linear Functionals and Projections on the Weighted Spaces of Analytic Functions

Under some general assumptions on weight, we give a complete characterization of the Cauchy transform of continuous linear functionals on the weighted spaces of holomorphic functions in a ball. We construct an integral projection that sends the weighted spaces of measurable and n-harmonic functions in the ball onto the corresponding spaces of holomorphic functions.     

A study of the effect of the loss function on Bayes Estimate,posterior risk and hazard function for Lindley distribution

In this paper, mathematical properties of Lindley distribution via Bayesian approach are derived under different loss functions. These properties include: Bayes Estimators, posterior risks and failure rate function for simulation scheme. Elicitation of hyperparameters is also discussed. A real life application to waiting time data at the bank is also described for the developed procedures (also using WinBUGS). Results are compared on the basis of posterior risk.     

金融市场波动性CARR类模型与GARCH类模型的比较研究

在CARR模型基础上提出它的衍生模型ABSCARR模型,并利用广义误差分布(GED)讨论了它们的条件残差分布问题,最后运用CARR类模型对高频金融时间序列进行了实证分析.研究结果表明:CARR及其衍生模型在高频金融时间序列的价格波动性捕捉方面具有良好的效果,而GED的引入可以很好的用于分析CARR模型具体的条件分布情况,而CARR模型的条件残差分布应该并非只有指数分布与威布尔分布两种形式.     

The effect of replenishment policies on the bullwhip effect: A transfer function approach

An important phenomenon in supply chain management, known as the bullwhip effect, suggests that demand variability increases as one moves up a supply chain. This paper examines the influence of different replenishment policies on the occurrence of the bullwhip effect. The paper demonstrates that certain replenishment policies can in themselves be inducers of the bullwhip effect, while others inherently lower demand variability. The main causes of increase in variability are projections of future demand expectations, which result in over-exaggerated responses to changes in demand. We suggest that through appropriate selection and use of certain replenishment rules, the bullwhip effect can be avoided, subsequently allowing supply chain management costs to be lowered.     

关于精确罚函数和线性规划的一个注记

本文给出了线性规划（LP）的解和其相应的精确罚函数解的等价性,证明是初等的,简明的。