Design of an S Function for Robust Regression Using Iteratively Reweighted Least Squares

Abstract

We develop a set of 5 functions for robust regression using the technique of iteratively reweighted least squares (IRLS). Together with a set of weight functions, function rreg is simple to understand and provides great flexibility for IRLS methods. This article focuses on the programming strategies adopted to achieve the twin goals of power and simplicity.     

A constrained-optimization based half-quadratic algorithm for robustly fitting sets of linearly parametrized curves

We consider the problem of multiple fitting of linearly parametrized curves, that arises in many computer vision problems such as road scene analysis. Data extracted from images usually contain non-Gaussian noise and outliers, which makes classical estimation methods ineffective. In this paper, we first introduce a family of robust probability density functions which appears to be well-suited to many real-world problems. Also, such noise models are suitable for defining continuation heuristics to escape shallow local minima and their robustness is devised in terms of breakdown point. Second, the usual Iterative Reweighted Least Squares (IRLS) robust estimator is extended to the problem of robustly estimating sets of linearly parametrized curves. The resulting, non-convex optimization problem is tackled within a Lagrangian approach, leading to the so-called Simultaneous Robust Multiple Fitting (SRMF) algorithm, whose global convergence to a local minimum is proved using results from constrained optimization theory.     

The use of relative residues in linear regression of experimental data with errors in both fit variables

A new regression model which mininizes the sum of squares of relative residues for data with errors in both fit variables is presented for linear fits. Expressions are derived for the slope, intercept and their respective errors. A detailed comparison is made between the new improved relative least squares (IRLS) model and other linear regression models, using three sets of data points. It is shown that IRLS provides the best compromise between, respectively, quality of fit, and a realistic representation of the physical situation of errors in both fit variables.     

Efficient fourth orderP-stable formulae

In this paper, a family of fourth orderP-stable methods for solving second order initial value problems is considered. When applied to a nonlinear differential system, all the methods in the family give rise to a nonlinear system which may be solved using a modified Newton method. The classical methods of this type involve at least three (new) function evaluations per iteration (that is, they are 3-stage methods) and most involve using complex arithmetic in factorising their iteration matrix. We derive methods which require only two (new) function evaluations per iteration and for which the iteration matrix is a true real perfect square. This implies that real arithmetic will be used and that at most one real matrix must be factorised at each step. Also we consider various computational aspects such as local error estimation and a strategy for changing the step size.     

美式期权隐含波动率校准问题的研究

研究的是美式期权的隐含波动率校准问题.首先提出一个正则化的最小二乘方法,在对其惩罚问题研究后找到最小二乘问题的最优条件,并给出美式期权波动率校准问题的算法.最后,通过数值算例说明了方法的有效性.     

The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector

The nucleolus and the prenucleolus are solution concepts for TU games based on the excess vector that can be associated to any payoff vector. Here we explore some solution concepts resulting from a payoff vector selection based also on the excess vector but by means of an assessment of their relative fairness different from that given by the lexicographical order. We take the departure consisting of choosing the payoff vector which minimizes the variance of the resulting excesses of the coalitions. This procedure yields two interesting solution concepts, both a prenucleolus-like and a nucleolus-like notion, depending on which set is chosen to set up the minimizing problem: the set of efficient payoff vectors or the set of inputations. These solution concepts, which, paralleling the prenucleolus and the nucleolus, we call least square prenucleolus and least square nucleolus, are easy to calculate and exhibit nice properties. Different axiomatic characterizations of the former are established, some of them by means of consistency for a reasonable reduced game concept.     

On a problem of Diophantus for rationals

Let q be a nonzero rational number. We investigate for which q there are infinitely many sets consisting of five nonzero rational numbers such that the product of any two of them plus q is a square of a rational number. We show that there are infinitely many square-free such q and on assuming the Parity Conjecture for the twists of an explicitly given elliptic curve we derive that the density of such q is at least one half. For the proof we consider a related question for polynomials with integral coefficients. We prove that, up to certain admissible transformations, there is precisely one set of non-constant linear polynomials such that the product of any two of them except one combination, plus a given linear polynomial is a perfect square.     

On-line secret sharing

In a perfect secret sharing scheme the dealer distributes shares to participants so that qualified subsets can recover the secret, while unqualified subsets have no information on the secret. In an on-line secret sharing scheme the dealer assigns shares in the order the participants show up, knowing only those qualified subsets whose all members she has seen. We often assume that the overall access structure (the set of minimal qualified subsets) is known and only the order of the participants is unknown. On-line secret sharing is a useful primitive when the set of participants grows in time, and redistributing the secret when a new participant shows up is too expensive. In this paper we start the investigation of unconditionally secure on-line secret sharing schemes. The complexity of a secret sharing scheme is the size of the largest share a single participant can receive over the size of the secret. The infimum of this amount in the on-line or off-line setting is the on-line or off-line complexity of the access structure, respectively. For paths on at most five vertices and cycles on at most six vertices the on-line and offline complexities are equal, while for other paths and cycles these values differ. We show that the gap between these values can be arbitrarily large even for graph based access structures. We present a general on-line secret sharing scheme that we call first-fit. Its complexity is the maximal degree of the access structure. We show, however, that this on-line scheme is never optimal: the on-line complexity is always strictly less than the maximal degree. On the other hand, we give examples where the first-fit scheme is almost optimal, namely, the on-line complexity can be arbitrarily close to the maximal degree. The performance ratio is the ratio of the on-line and off-line complexities of the same access structure. We show that for graphs the performance ratio is smaller than the number of vertices, and for an infinite family of graphs the performance ratio is at least constant times the square root of the number of vertices.     

多元广义岭估计及K值选取的 Q(c)准则

当自变量间存在复共线性时,最小二乘估计就表现出不稳定并可能导致错误的结果.本文采用广义岭估计β(K)来估计多元线性模型的回归系数β=vec(B),通过岭参数K值的选取,可使广义岭估计的均方误差MSE小于最小二乘估计的MSE.指出了广义岭估计中根据MSE准则选取K值存在的主要缺陷,采用了一种选取K值的新准则Q(c),它包含MSE准则和最小二乘LS准则作为特例,从理论上证明和讨论了Q(c)准则的优良性,阐明了c值的统计含义,并给出了确定c值的方法.     

Comparison of numerical integration formulas

In this paper we study the mean square error of numerical integration, when the integrand is a random stationary process. We obtain exact asymptotic errors of classical quadrature formulas and give lower and upper bounds for the least mean square error.     

Weakly directed self-avoiding walks

We define a new family of self-avoiding walks (SAW) on the square lattice, called weakly directed walks. These walks have a simple characterization in terms of the irreducible bridges that compose them. We determine their generating function. This series has a complex singularity structure and in particular, is not D-finite. The growth constant is approximately 2.54 and is thus larger than that of all natural families of SAW enumerated so far (but smaller than that of general SAW, which is about 2.64). We also prove that the end-to-end distance of weakly directed walks grows linearly. Finally, we study a diagonal variant of this model.     

Second variation of volume and energy of vector fields. Stability of Hopf vector fields

In this paper we compute the Hessian of the volume of unit vector fields at a minimal one. We also find the Hessians of a family of functionals thus generalizing the known results concerning second variation of the energy or total bending. We use them to study the stability of Hopf vector fields on and to show that they are stable for , but that for there is such that for the index is at least . Received May 10, 1999 / Published online April 12, 2001     

Logical Squares for Classical Logic Sentences

In this paper, with reference to relationships of the traditional square of opposition, we establish all the relations of the square of opposition between complex sentences built from the 16 binary and four unary propositional connectives of the classical propositional calculus (CPC). We illustrate them by means of many squares of opposition and, corresponding to them—octagons, hexagons or other geometrical objects.     

Estimating an even spherical measure from its sine transform

To reconstruct an even Borel measure on the unit sphere from finitely many values of its sine transform a least square estimator is proposed. Applying results by Gardner, Kiderlen and Milanfar we estimate its rate of convergence and prove strong consistency. We close this paper by giving an estimator for the directional distribution of certain threedimensional stationary Poisson processes of convex cylinders which have applications in material science. When writing this paper the author was funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

The period function of generalized Loudʼs centers

In this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied. This family is an extension to higher degree of Loud?s systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. We study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center. Moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period, giving also its character (maximum or minimum). Finally we propose a complete conjectural bifurcation diagram of the period function of these generalized Loud?s centers.     

A new family of regular semivalues and applications

We define a new family of values for cooperative games, including as a particular case the Shapley value. They are defined on the collection of the unanimity games, then extended by linearity. Our most relevant result shows that the family of the weighting coefficients characterizing the values so defined is an open curve on the simplex of the regular semivalues. We give an explicit formula for the values when the parameter characterizing the family is a natural number and we offer an algorithm to calculate them in weighted majority games, slightly extending previous results (see Bilbao et al., TOP, 8:191–213, 2000). The paper ends with two applications. The first one is classical, and serves to see how the indices behave with respect to the Shapley and Banzhaf values in the case of the EU parliament and in the UN Security Council. The second one is much more recent: it deals with the microarray games, introduced in Moretti et al. (TOP, 15:256–280, 2007), which are average of unanimity games. The idea is to rank genes taken from DNA of patients affected by a specific disease, with the aim of singling out a group of genes potentially responsible of the disease. In this last case we consider some microarray data available on the net and concerning some specific diseases and we show that several genes mentioned in the medical literature as potentially responsible for the onset of the disease are present in the first places according to our rankings.     

Accelerate overrelaxation methods for rank deficient linear systems

In this paper, we apply accelerated overrelaxation (AOR) methods to find the least square solution of minimal norm to the linear system

Ay=b

where is a matrix of rank r and . We first augment the system to a block 4×4 consistent system, and then split the augmented coefficient matrix by AOR subproper splitting. Intervals for the two relaxation parameters where the AOR iteration matrix is semiconvergent are presented. Also, we provide a method to compute the least square solution of minimal norm to the system.     

The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t,m,s)-nets over which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order for any ɛ > 0, where 2 ^m is the number of points. A similar result for lattice rules has previously been shown by Hickernell. Ligia L. Cristea is supported by the Austrian Research Fund (FWF), Project P 17022-N 12 and Project S 9609. Josef Dick is supported by the Australian Research Council under its Center of Excellence Program. Gunther Leobacher is supported by the Austrian Research Fund (FWF), Project S 8305. Friedrich Pillichshammer is supported by the Austrian Research Fund (FWF), Project P 17022-N 12, Project S 8305 and Project S 9609.     

回归系数Stein压缩估计的小样本性质

本文在广义均方误差（GMSE）准则下给出了回归系数β的Stein估计优于最小二乘（LS）估计的充分必要条件，然后在Pitman Closeness(PC)准则下比较了Stein估计相对于LS估计的优良性，本文最后给出了一个特别的注记。     

The Lagrange trigonometric interpolation polynomial with the minimal norm considered as an operator from <Emphasis Type="Italic">C</Emphasis><Subscript>2π</Subscript> to <Emphasis Type="Italic">C</Emphasis><Subscript>2π</Subscript>

In this paper we perform a comparative analysis of Lebesgue functions and constants of a family of Lagrange polynomials. We prove that if a polynomial from the family has the minimal norm in the space of square summable functions, then it also has the minimal norm as an operator which maps the space of continuous functions into itself.