Consistency of kernel‐based quantile regression

Quantile regression is used in many areas of applied research and business. Examples are actuarial, financial or biometrical applications. We show that a non‐parametric generalization of quantile regression based on kernels shares with support vector machines the property of consistency to the Bayes risk. We further use this consistency to prove that the non‐parametric generalization approximates the conditional quantile function which gives the mathematical justification for kernel‐based quantile regression. Copyright © 2008 John Wiley & Sons, Ltd.     

Call center service process analysis: Bayesian parametric and semi-parametric mixture modeling

The main focus of the call center research has been on models that assume all input distributions are known in queuing theory which gives birth to staffing and the estimation of operating characteristics. Studies investigating uncertainty of the input distributions and its implications on call center management are scarce. This study attempts to fill this gap by analyzing the call center service distribution behavior by using Bayesian parametric and semi-parametric mixture models that are capable of exhibiting non-standard behavior such as multi-modality, skewness and excess kurtosis motivated by real call center data. The study is motivated by the observation that different customer profiles might require different agent skill sets which can create additional sources of uncertainty in the behavior of service distributions. In estimating model parameters, Markov chain Monte Carlo methods such as the Gibbs sampler and the reversible jump algorithms are presented and the implications of using such models on system performance and staffing are discussed.     

An evaluation of the multivariate dispersion charts with estimated parameters under non‐normality

Various charts such as |S|, W, and G are used for monitoring process dispersion. Most of these charts are based on the normality assumption, while exact distribution of the control statistic is unknown, and thus limiting distribution of control statistic is employed which is applicable for large sample sizes. In practice, the normality assumption of distribution might be violated, while it is not always possible to collect large sample size. Furthermore, to use control charts in practice, the in‐control state usually has to be estimated. Such estimation has a negative effect on the performance of control chart. Non‐parametric bootstrap control charts can be considered as an alternative when the distribution is unknown or a collection of large sample size is not possible or the process parameters are estimated from a Phase I data set. In this paper, non‐parametric bootstrap multivariate control charts |S|, W, and G are introduced, and their performances are compared against Shewhart‐type control charts. The proposed method is based on bootstrapping the data used for estimating the in‐control state. Simulation results show satisfactory performance for the bootstrap control charts. Ultimately, the proposed control charts are applied to a real case study.     

On the stability of retrial queues

We consider the following Type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the call (customer) waits in line, and will be served according to the FIFO order. If λ is the arrival rate (average number per time unit) of calls and μ is one over the expected service time in the facility, it is well known that μ > λ is not always sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition. In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent i.i.d. sequences and the retrial times are exponentially distributed; (ii) we establish conditions for strong coupling convergence to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival and retrial times are i.i.d. exponentially distributed; or when inter-arrival times are general stationary ergodic, and service and retrial times are i.i.d. exponentially distributed; (iii) we obtain conditions for the existence of uniform exponential bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions for boundedness in distribution for the case of nonpatient (or non persistent) customers. This revised version was published online in June 2006 with corrections to the Cover Date.     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.     

Weekday dependence of German stock market returns

The so‐called ‘Monday effect’ has been found for various stock markets of the world. The empirical finding that Monday returns are significantly smaller than returns measured for the remaining days of the week calls the efficiency hypothesis for pricing processes operating on stock markets into question. Investigating an index series measured at the Frankfurt stock exchange the paper compares estimation results of parametric and non‐parametric autoregressive models with respect to possible weekday dependence of return data. Allowing for heteroskedastic error distributions the wild bootstrap is used to infer against time‐varying means and correlation of return data in parametric models and to obtain confidence bands for non‐parametric estimates. It is shown that time dependence is an important feature describing the dynamics of German stock market returns in the period 1960–1979. Within two subsamples obtained from the period 1980–1997 the evidence in favour of such effects is mitigated substantially. Copyright © 2000 John Wiley & Sons, Ltd.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

Two thresholds of a batch arrival queueing system under modified T vacation policy with startup and closedown

This paper studies the operating characteristics of the variant of an M^[x]/G/1 vacation queue with startup and closedown times. After all the customers are served in the system exhaustively, the server shuts down (deactivates) by a closedown time, and then takes at most J vacations of constant time length T repeatedly until at least one customer is found waiting in the queue upon returning from a vacation. If at least one customer is present in the system when the server returns from a vacation, then the server reactivates and requires a startup time before providing the service. On the other hand, if no customers arrive by the end of the J th vacation, the server remains dormant in the system until at least one customer arrives. We will call the vacation policy modified T vacation policy. We derive the steady‐state probability distribution of the system size and the queue waiting time. Other system characteristics are also investigated. The long‐run average cost function per unit time is developed to determine the suitable thresholds of T and J that yield a minimum cost. Copyright © 2007 John Wiley & Sons, Ltd.     

A MCMC approach for modeling customer lifetime behavior using the COM‐Poisson distribution

One of the major challenges associated with the measurement of customer lifetime value is selecting an appropriate model for predicting customer future transactions. Among such models, the Pareto/negative binomial distribution (Pareto/NBD) is the most prevalent in noncontractual relationships characterized by latent customer defections; ie, defections are not observed by the firm when they happen. However, this model and its applications have some shortcomings. Firstly, a methodological shortcoming is that the Pareto/NBD, like all lifetime transaction models based on statistical distributions, assumes that the number of transactions by a customer follows a Poisson distribution. However, many applications have an empirical distribution that does not fit a Poisson model. Secondly, a computational concern is that the implementation of Pareto/NBD model presents some estimation challenges specifically related to the numerous evaluation of the Gaussian hypergeometric function. Finally, the model provides 4 parameters as output, which is insufficient to link the individual purchasing behavior to socio‐demographic information and to predict the behavior of new customers. In this paper, we model a customer's lifetime transactions using the Conway‐Maxwell‐Poisson distribution, which is a generalization of the Poisson distribution, offering more flexibility and a better fit to real‐world discrete data. To estimate parameters, we propose a Markov chain Monte Carlo algorithm, which is easy to implement. Use of this Bayesian paradigm provides individual customer estimates, which help link purchase behavior to socio‐demographic characteristics and an opportunity to target individual customers.     

Single line queue with repeated demands

We analyze a model of a queueing system in which customers can only call in to request service: if the server is free, the customer enters service immediately, but if the service system is occupied, the unsatisfied customer must break contact and reinitiate his request later. Such a customer is said to be in “orbit”. In this paper we consider three models characterized by the discipline governing the order of re-request of service from orbit. First, all customers in orbit can reapply, but are discouraged and reduce their rate of demand as more customers join the orbit. Secondly, the FCFS discipline operates for the unsatisfied customers in orbit. Finally, the LCFS discipline governs the customers in orbit and the server takes an exponentially distributed vacation after each service is completed. We calculate several characteristics quantities of such systems, assuming a general service-time distribution and different exponential distributions for the times between arrivals of first and repeat requests.     

Minimizing a general loss function in off‐line quality control

We consider in the present paper the analysis of parameter designs in off‐line quality control. The main objective is to seek levels of the production factors that would minimize the expected loss. Unlike classical analyses which focus on the analysis of the mean and variance in minimizing a quadratic loss function, the proposed method is applicable to a general loss function. An appropriate transformation is first sought to eliminate the dependency of the variance on the mean (to achieve ‘separation’ in the terminology of Box). This is accomplished through a preliminary analysis using a recently proposed parametric heteroscedastic regression model. With the dependency of the variance on the mean eliminated, methods with established properties can be applied to estimate simultaneously the mean and the variance functions in the new metric. The expected loss function is then estimated and minimized based on a distributional free procedure using the empirical distribution of the standardized residuals. This alleviates the need for a full parametric model, which, if incorrectly specified, may lead to biased results. Although a transformation is employed as an intermediate step of analysis, the loss function is minimized in its original metric. Copyright © 2002 John Wiley & Sons, Ltd.     

Sequencing and scheduling N customers for a stochastic server

In this paper, we study the problem of sequencing and scheduling N customers for a single-server system. The goal is to balance the expected customer flow times and the expected system completion time. Customers are scheduled to enter the system by appointments only and the service times are exponentially distributed with different rates. The optimization of such a system involves determining the customer service order (sequencing) and the interarrival times (scheduling). We show that the service order depends upon the order of service rates and the optimal schedule can be obtained by solving a set of nonlinear equations. Numerical examples are used to illustrate the method.     

Real‐time road traffic forecasting using regime‐switching space‐time models and adaptive LASSO

Smart transportation technologies require real‐time traffic prediction to be both fast and scalable to full urban networks. We discuss a method that is able to meet this challenge while accounting for nonlinear traffic dynamics and space‐time dependencies of traffic variables. Nonlinearity is taken into account by a union of non‐overlapping linear regimes characterized by a sequence of temporal thresholds. In each regime, for each measurement location, a penalized estimation scheme, namely the adaptive absolute shrinkage and selection operator (LASSO), is implemented to perform model selection and coefficient estimation simultaneously. Both the robust to outliers least absolute deviation estimates and conventional LASSO estimates are considered. The methodology is illustrated on 5‐minute average speed data from three highway networks. Copyright © 2012 John Wiley & Sons, Ltd.     

Signal moments for the short‐time Fourier transform associated with Hardy–Sobolev derivatives

The short‐time Fourier transform has been shown to be a powerful tool for non‐stationary signals and time‐varying systems. This paper investigates the signal moments in the Hardy–Sobolev space that do not usually have classical derivatives. That is, signal moments become valid for non‐smooth signals if we replace the classical derivatives by the Hardy–Sobolev derivatives. Our work is based on the extension of Cohen's contributions to the local and global behaviors of the signal. The relationship of the moments and spreads of the signal in the time, frequency and short‐time Fourier domain are established in the Hardy–Sobolev space. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential state estimator design for discrete‐time neural networks with discrete and distributed time‐varying delays

In this article, the probl of state estimation for discrete‐time neural networks with mixed time‐varying delays is investigated. The mixed time delays consist of both discrete and distributed delays. An appropriate Lyapunov–Krasovskii functional put forward to reflect the mixed time‐varying delays is proposed to establish sufficient conditions for the existence of admissible state estimators. The conditions are described in the form of linear matrix inequalities (LMIs), which guarantee the estimation error to be globally exponentially stable in the presence of mixed time‐varying delays. Then, the desired estimator matrix gain can be characterized in terms of the solution to these LMIs. A numerical example is addressed to show the effectiveness of the proposed design method. © 2014 Wiley Periodicals, Inc. Complexity 20: 38–48, 2014     

IPO estimation of heaviness of the distribution beyond regularly varying tails

Abstract

We introduce a completely novel method for estimation of the parameter which governs the tail behavior of the cumulative distribution function of the observed random variable. We call it Inverse Probabilities for p-Outside values (IPO) estimation method. We show that this approach is applicable for wider class of distributions than the one with regularly varying tails. We demonstrate that IPO method is a valuable competitor to regularly varying tails based estimation methods. Some of the properties of the estimators are derived. The results are illustrated by a convenient simulation study.     

Parameter identification based on FE‐models by genetic algorithms

A realistic and reliable model is an important precondition for the simulation of revitalization tasks as well as for the estimation of properties of existing buildings. Within one theory the parameters of the model should be approximated best by gradually performed experiments and their analysis. Usually this kind of optimization problems leads into non‐convex non‐differentiable objective function spaces with high dimensions. Normally ore complex structures are modeled using finite element method. We present a method of identifying Young's modulus for a beam and a plate by using FE‐models and genetic optimization algorithms for parameter identification.     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

Blended kernel approximation in the ℋ︁‐matrix techniques

Several types of ??‐matrices were shown to provide a data‐sparse approximation of non‐local (integral) operators in FEM and BEM applications. The general construction is applied to the operators with asymptotically smooth kernel function provided that the Galerkin ansatz space has a hierarchical structure. The new class of ??‐matrices is based on the so‐called blended FE and polynomial approximations of the kernel function and leads to matrix blocks with a tensor‐product of block‐Toeplitz (block‐circulant) and rank‐k matrices. This requires the translation (rotation) invariance of the kernel combined with the corresponding tensor‐product grids. The approach allows for the fast evaluation of volume/boundary integral operators with possibly non‐smooth kernels defined on canonical domains/manifolds in the FEM/BEM applications. (Here and in the following, we call domains canonical if they are obtained by translation or rotation of one of their parts, e.g. parallelepiped, cylinder, sphere, etc.) In particular, we provide the error and complexity analysis for blended expansions to the Helmholtz kernel. Copyright © 2002 John Wiley & Sons, Ltd.     

Local and non‐local boundary quenching

The quenching problem is examined for a one‐dimensional heat equation with a non‐linear boundary condition that is of either local or non‐local type. Sufficient conditions are derived that establish both quenching and non‐quenching behaviour. The growth rate of the solution near quenching is also given for a power‐law non‐linearity. The analysis is conducted in the context of a nonlinear Volterra integral equation that is equivalent to the initial–boundary value problem. Copyright © 1999 John Wiley & Sons, Ltd.