On detection of change points using mean vectors

In this paper, the authors consider the problem of change points within the framework of model selection and propose a procedure for estimating the locations of change points when the number of change points is known. The strong consistency of this procedure is also established. The problem of detecting change points is discussed within the framework of the simultaneous test procedure. The case where the number of change points is unknown will be discussed in another paper.This project is supported by the National Natural Science Foundation of China and by the Air Office of Scientific Research of the United States.     

A branch and bound algorithm for locating input and output points of departments on the block layout

We consider the problem of locating input and output (I/O) points of each department for a given layout. The objective of the problem is to minimise the total distance of material flows between the I/O points. Here, distances between the I/O points are computed as the lengths of the shortest path (not the rectilinear distances) between the I/O points. We developed a procedure to eliminate dominated candidate positions of I/O points that do not need to be considered. With this procedure, a large number of dominated candidate positions can be eliminated. A linear programming (LP) model for minimising the total rectilinear distance of flows is used to obtain a lower bound. Using the elimination procedure and the LP model, a branch and bound algorithm is developed to find an optimal location of the I/O points. Results from computational experiments show that the suggested algorithm finds optimal solutions in a very short time even for large-sized problems.     

Using the pointwise dimension to filter time series

This paper provides an insight into dimension analysis from time series. In particular, we propose a procedure based on the pointwise dimension in order to extract, for each embedding dimension, the subset of points in the phase space (and the corresponding ones in the time series) which give rise to the scaling behaviour. We may consider the output time series as the result of a filtering process, based on correlations of points in the phase-space domain. Furthermore, the procedure gives the statistics of points which determine the scaling behaviour.     

Limit points of the iterative scaling procedure

The iterative scaling procedure (ISP) is an algorithm which computes a sequence of matrices, starting from some given matrix. The objective is to find a matrix ’proportional’ to the given matrix, having given row and column sums. In many cases, for example if the initial matrix is strictly positive, the sequence is convergent. It is known that the sequence has at most two limit points. When these are distinct, convergence to these two points can be slow. We give an efficient algorithm which finds the limit points, invoking the ISP only on subproblems for which the procedure is convergent.     

Restructuring forward step of MARS algorithm using a new knot selection procedure based on a mapping approach

In high dimensional data modeling, Multivariate Adaptive Regression Splines (MARS) is a popular nonparametric regression technique used to define the nonlinear relationship between a response variable and the predictors with the help of splines. MARS uses piecewise linear functions for local fit and apply an adaptive procedure to select the number and location of breaking points (called knots). The function estimation is basically generated via a two-stepwise procedure: forward selection and backward elimination. In the first step, a large number of local fits is obtained by selecting large number of knots via a lack-of-fit criteria; and in the latter one, the least contributing local fits or knots are removed. In conventional adaptive spline procedure, knots are selected from a set of all distinct data points that makes the forward selection procedure computationally expensive and leads to high local variance. To avoid this drawback, it is possible to restrict the knot points to a subset of data points. In this context, a new method is proposed for knot selection which bases on a mapping approach like self organizing maps. By this method, less but more representative data points are become eligible to be used as knots for function estimation in forward step of MARS. The proposed method is applied to many simulated and real datasets, and the results show that it proposes a time efficient forward step for the knot selection and model estimation without degrading the model accuracy and prediction performance.     

An Interactive MOLP Procedure for the Extension of DEA to Effectiveness Analysis

This paper presents a new, interactive multi-objective linear-programming procedure to aid decision-makers in setting up goals for desired outputs. The procedure relies on empirical production functions generated by the use of data envelopment analysis. It presents the decision-maker with a set of alternative efficient points in order either to compare different sets of inputs in terms of their effectiveness for goal achievement, or to set goals against which future management performance may be measured. With each iteration the new information provided by the decision-maker is used to adjust the procedure, leading to points which have greater effectiveness utility for the decision-maker. A numerical example is provided along with guidelines for future applications.     

A method for generating uniformly distributed points onN-dimensional spheres

Summary In this note a computer procedure to transform uniform random variables into random points uniformly distributed on an N-dimensional sphere is presented. The procedure is much simpler than the ones thus far published.     

A method for the computation of nonsimple turning points corresponding to cusps

A direct method is described for the approximation of nonsimple turning points, corresponding to cusp points, of nonlinear operator equations depending on two parameters. The procedure is based on the application of a special projection method to the computation of simple turning points of a suitable augmented system. Numerical examples illustrate the features of the proposed algorithm.     

Boosting the feasibility pump

The feasibility pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP infeasible solutions. The process attempts to minimize the distance between consecutive iterates, producing an integer feasible solution when closing the distance between them. We investigate the benefits of enhancing the rounding procedure with a clever integer line search that efficiently explores a large set of integer points. An extensive computational study on benchmark instances demonstrates the efficacy of the proposed approach.     

Degenerating the elliptic Schlesinger system

We study various ways of degenerating the Schlesinger system on the elliptic curve with R marked points. We construct a limit procedure based on an infinite shift of the elliptic curve parameter and on shifts of the marked points. We show that using this procedure allows obtaining a nonautonomous Hamiltonian system describing the Toda chain with additional spin sl(N, ?) degrees of freedom.     

The approximation of generalized turning points by projection methods with superconvergence to the critical parameter

Summary A procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between Banach spaces. This characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization. Applications are given to parameter dependent two-point boundary value problems. In particular, collocation at Gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points.     

Stress recovery procedure for solving boundary value problems in the mechanics of a deformable solid by the finite element method

A stress recovery procedure, based on the determination of the forces at the mesh points using a stiffness matrix obtained by the finite element method for the variational Lagrange equation, is described. The vectors of the forces reduced to the mesh points are constructed for the known stiffness matrices of the elements using the displacements at the mesh points found from the solution of the problem. On the other hand, these mesh point forces are determined in terms of the unknown forces distributed over the surface of an element and given shape functions. As a result, a system of Fredholm integral equations of the first kind is obtained, the solution of which gives these distributed forces. The stresses at the mesh points are determined for the values of these forces found on the surfaces of the finite element mesh (including at the mesh points) using the Cauchy relations, which relate the forces, stresses and the normal to the surface. The special features of the use of the stress recovery procedure are demonstrated for a plane problem in the linear theory of elasticity.     

An algorithm for numerical differentiation of a function of one real variable

An Algol 60 procedure is described which will estimate the first, second or third order derivative of a function f(x) at a point x₀, using polynomial interpolation to values of f(x) at points on the real axis. This procedure does not require the user to provide information about the accuracy of the function values. The procedure itself monitors the noise level in these values, and allows for the effect of noise on the estimated derivative. The evaluation points may be restricted to one side of x₀ if f(x) makes this necessary, and are chosen either to minimise the error in the result, or to achieve a specified tolerance with as few function evaluations as possible.     

Construction of a sphere by using projective Bézier patches

A projective (rational) rectangular second-order Bézier patch is determined by nine points in 3-space and by positive weights assigned to these points. The shape of the patch can be changed by varying the positions of the control points and the weight values. A procedure for constructing a sphere by using a projective rectangular second-order Bézier patch is suggested. A special set of control points, three of which are multiple, and weights is used.     

A Cutting-Point Technique For Solving Nonlinear State Regulator Problems

In this paper, we present a technique for solving an optimizationproblem of Lagrange type where the system equations are stiffand the system can be expressed explicitly in terms of a singular-perturbationparameter. Such problems arise in control theory and in otherapplication areas. The proposed procedure consists of solvingthe singularly perturbed two-point boundary-value problem comprisedby the coupled state and adjoint equations arising from thefirst-order necessary conditions for optimality. It is assumedthat the coupled system has no turning points, and the solutionis accomplished by making an analytic stretching in the boundarylayers to give three explicit boundary-value problems whichare treated separately. The novel feature of the procedure isthat the explicit boundary conditions for each problem are obtained,at the selected cut points, from the solution to the reducedproblem. The application of the procedure is described for twoexamples.     

Symmetrized-lexicographic order and the method of points

A variant of lexicographic order called symmetrized-lexicographic order is defined. The symmetrized-lexicographic order finds its application in the goal programming procedure called the method of points. The symmetrized-lexicographic order is shown to be representable using linear algebra and, thus, the method of points can be implemented as a linear programming problem.     

Smooth curve interpolation

In a recent paper [1] a method was presented for smooth curve interpolation by means of minimising the energy of an idealised elastic beam constrained to pass through given data points. This method of obtaining the curve required an enormous amount of computation, since it involved the repeated solution of a fourth order differential equation and the minimisation of a function of many variables, the number of variables being equal to the number of data points. In this paper a much simpler method is presented for finding the minimum energy curve in which the minimisation procedure is eliminated. An ALGOL procedure to perform the curve fit is supplied.     

Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method

Spectral discretization methods are well established methods for the computation of characteristic roots of time-delay systems. In this paper a method is presented for computing all characteristic roots in a given right half plane. In particular, a procedure for the automatic selection of the number of discretization points is described. This procedure is grounded in the connection between a spectral discretization and a rational approximation of exponential functions. First, a region that contains all desired characteristic roots is estimated. Second, the number of discretization points is selected in such a way that in this region the rational approximation of the exponential functions is accurate. Finally, the characteristic roots approximations, obtained from solving the discretized eigenvalue problem, are corrected up to the desired precision by a local method. The effectiveness and robustness of the procedure are illustrated with several examples and compared with DDE-BIFTOOL.     

Spline Estimation of Discontinuous Regression Functions

Abstract

This article deals with regression function estimation when the regression function is smooth at all but a finite number of points. An important question is: How can one produce discontinuous output without knowledge of the location of discontinuity points? Unlike most commonly used smoothers that tend to blur discontinuity in the data, we need to find a smoother that can detect such discontinuity. In this article, linear splines are used to estimate discontinuous regression functions. A procedure of knot-merging is introduced for the estimation of regression functions near discontinuous points. The basic idea is to use multiple knots for spline estimates. We use an automatic procedure involving the least squares method, stepwise knot addition, stepwise basis deletion, knot-merging, and the Bayes information criterion to select the final model. The proposed method can produce discontinuous outputs. Numerical examples using both simulated and real data are given to illustrate the performance of the proposed method.     

Solving a bicriterion scheduling problem

Consider n jobs to be sequenced on a single machine. The objective functions to be minimized are the holding cost and the maximum tardiness. We first characterize the set of efficient points and then proceed to give a pseudo-polynomial algorithm to enumerate all these efficient points. Computational results illustrate the usefulness of the procedure.