Optimal training sample allocation and asymptotic expansions for error rates in discriminant analysis

The sample-based rule obtained from Bayes classification rule by replacing the unknown parameters by ML estimates from a stratified training sample is used for the classification of a random observationX into one ofL populations. The asymptotic expansions in terms of the inverses of the training sample sizes for cross-validation, apparent and plug-in error rates are found. These are used to compare estimation methods of the error rate for a wide range of regular distributions as probability models for considered populations. The optimal training sample allocation minimizing the asymptotic expected error regret is found in the cases of widely applicable, positively skewed distributions (Rayleigh and Maxwell distributions). These probability models for populations are often met in ecology and biology. The results indicate that equal training sample sizes for each populations sometimes are not optimal, even when prior probabilities of populations are equal.     

Stable Grothendieck polynomials and <Emphasis Type="Italic">K</Emphasis>-theoretic factor sequences

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of Fomin and Kirillov (Proc Formal Power Series Alg Comb, 1994) in the basis of stable Grothendieck polynomials for partitions. This gives a common generalization, as well as new proofs of the rule of Fomin and Greene (Discret Math 193:565–596, 1998) for the expansion of the stable Schubert polynomials into Schur polynomials, and the K-theoretic Grassmannian Littlewood–Richardson rule of Buch (Acta Math 189(1):37–78, 2002). The proof is based on a generalization of the Robinson–Schensted and Edelman–Greene insertion algorithms. Our results are applied to prove a number of new formulas and properties for K-theoretic quiver polynomials, and the Grothendieck polynomials of Lascoux and Schützenberger (C R Acad Sci Paris Ser I Math 294(13):447–450, 1982). In particular, we provide the first K-theoretic analogue of the factor sequence formula of Buch and Fulton (Invent Math 135(3):665–687, 1999) for the cohomological quiver polynomials.     

Skew quantum Murnaghan–Nakayama rule

We extend recent results of Assaf and McNamara on a skew Pieri rule and a skew Murnaghan–Nakayama rule to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf–McNamara’s original proof, and one via Lam–Lauve–Sotille’s skew Littlewood–Richardson rule. We end with some conjectures for skew rules for Hall–Littlewood polynomials.     

Bayes estimates and classification problem for chi-squared and Wishart's distributions

Bayes estimators are proposed for the likelihood functions of random matrices having Wishart's distribution. These estimators are used to construct an asymptotically optimal classification rule. The classification problem in the case of the chi-squared distribution is also considered. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 11–18, Perm, 1990.     

Influence of observations on the misclassification probability in quadratic discriminant analysis

In this paper it is studied how observations in the training sample affect the misclassification probability of a quadratic discriminant rule. An approach based on partial influence functions is followed. It allows to quantify the effect of observations in the training sample on the performance of the associated classification rule. Focus is on the effect of outliers on the misclassification rate, merely than on the estimates of the parameters of the quadratic discriminant rule. The expression for the partial influence function is then used to construct a diagnostic tool for detecting influential observations. Applications on real data sets are provided.     

Classification of (0,2)-geometries embedded in AG (3,q)

In De Clerck and Delanote (Des. Codes Cryptogr, 32: 103–110, 2004) it is shown that if a (0,α)-geometry with α ≥  3 is fully embedded in AG (n,q) then it is a linear representation. In De Feyter (J. Combin Theory Ser A, 109(1): 1–23, 2005; Discrete math, 292: 45–54, 2005) the (0,2)-geometries fully embedded in AG(3,q) are classified apart from two open cases. In this paper, we solve these two open cases. This classification for AG(3,q) is used in De Feyter (Adv Geom, 5: 279–292, 2005) to classify the (0,2)-geometries fully embedded in AG(n,q).      

Minimal codewords in Reed–Muller codes

Minimal codewords were introduced by Massey (Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp 276–279, 1993) for cryptographical purposes. They are used in particular secret sharing schemes, to model the access structures. We study minimal codewords of weight smaller than 3 · 2 ^m−r in binary Reed–Muller codes RM(r, m) and translate our problem into a geometrical one, using a classification result of Kasami and Tokura (IEEE Trans Inf Theory 16:752–759, 1970) and Kasami et al. (Inf Control 30(4):380–395, 1976) on Boolean functions. In this geometrical setting, we calculate numbers of non-minimal codewords. So we obtain the number of minimal codewords in the cases where we have information about the weight distribution of the code RM(r, m). The presented results improve previous results obtained theoretically by Borissov et al. (Discrete Appl Math 128(1), 65–74, 2003), and computer aided results of Borissov and Manev (Serdica Math J 30(2-3), 303–324, 2004). This paper is in fact an extended abstract. Full proofs can be found on the arXiv.     

A combinatorial approach to the classification problem

We study the two-group classification problem which involves classifying an observation into one of two groups based on its attributes. The classification rule is a hyperplane which misclassifies the fewest number of observations in the training sample. Exact and heuristic algorithms for solving the problem are presented. Computational results confirm the efficiency of this approach.     

Bayes estimation and group classification for multivariate dependent observations

In this paper, a statistical decision rule based on the Bayes estimators for the group classification of dependent Gaussian observations is constructed. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 33–41. Perm, 1993.     

Asymptotic expansions for the statistic and risk function of a Bayesian classification rule

Asymptotic expansions for the stastistic and risk function of a Bayesian classification rule for the problem of assigning an observation to one of the two one-dimensional populations with unknown parameters are constructed. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 69–77, Perm, 1991.     

Estimators for the binomial distribution that dominate the MLE in terms of Kullback–Leibler risk

Estimators based on the mode are introduced and shown empirically to have smaller Kullback–Leibler risk than the maximum likelihood estimator. For one of these, the midpoint modal estimator (MME), we prove the Kullback–Leibler risk is below \frac12{\frac{1}{2}} while for the MLE the risk is above \frac12{\frac{1}{2}} for a wide range of success probabilities that approaches the unit interval as the sample size grows to infinity. The MME is related to the mean of Fisher’s Fiducial estimator and to the rule of succession for Jefferey’s noninformative prior.     

Estimation of dependences based on Bayesian correction of a committee of classification algorithms

Estimation of dependence of a scalar variable on the vector of independent variables based on a training sample is considered. No a priori conditions are imposed on the form of the function. An approach to the estimation of the functional dependence is proposed based on the solution of a finite number of special classification problems constructed on the basis of the training sample and on the subsequent prediction of the value of the function as a group decision. A statistical model and Bayes’ formula are used to combine the recognition results. A generic algorithm for constructing the regression is proposed for different approaches to the selection of the committee of classification algorithms and to the estimation of their probabilistic characteristics. Comparison results of the proposed approach with the results obtained using other models for the estimation of dependences are presented.     

Neural networks in mechatronics

An elementary introduction to the theory of artificial neural networks is given. Principles of their structural composition are presented. Methods for the neural-network training commonly used for different levels of intellectual control of mechatronic systems are formulated and substantiated. Neural-network approaches to typical problems of classification, digital signal processing, data compression, function interpolation and extrapolation, associative behavior, and optimization are stated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 81–103, 2005.     

On an equivariant estimate of the density of a matrix normal distribution

An optimal equivariant Bayes estimate of the density of a matrix normal distribution is obtained. This estimate is applied to the construction of the optimal Bayes group classification rule. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 29–39, Perm, 1990.     

Solving Differential-Algebraic Equations by Taylor Series (I): Computing Taylor Coefficients

This paper is one of a series underpinning the authors’ DAETS code for solving DAE initial value problems by Taylor series expansion. First, building on the second author’s structural analysis of DAEs (BIT, 41 (2001), pp. 364–394), it describes and justifies the method used in DAETS to compute Taylor coefficients (TCs) using automatic differentiation. The DAE may be fully implicit, nonlinear, and contain derivatives of order higher than one. Algorithmic details are given. Second, it proves that either the method succeeds in the sense of computing TCs of the local solution, or one of a number of detectable error conditions occurs. AMS subject classification (2000) 34A09, 65L80, 65L05, 41A58     

Fast Construction of the Fejér and Clenshaw–Curtis Quadrature Rules

We present an elegant algorithm for stably and quickly generating the weights of Fejér’s quadrature rules and of the Clenshaw–Curtis rule. The weights for an arbitrary number of nodes are obtained as the discrete Fourier transform of an explicitly defined vector of rational or algebraic numbers. Since these rules have the capability of forming nested families, some of them have gained renewed interest in connection with quadrature over multi-dimensional regions. AMS subject classification (2000) 65D32, 65T20, 65Y20     

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing e ^z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3^-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903^-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e ^z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D30, 41A20     

Decreasing serial cost sharing: an axiomatic characterization

The increasing serial cost sharing rule of Moulin and Shenker (Econometrica 60:1009–1037, 1992) and the decreasing serial rule of de Frutos (J Econ Theory 79:245–275, 1998) are known by their intuitive appeal and striking incentive properties. An axiomatic characterization of the increasing serial rule was provided by Moulin and Shenker (J Econ Theory 64:178–201, 1994). This paper gives an axiomatic characterization of the decreasing serial rule.     

Analysis of the consistency of a mixed integer programming-based multi-category constrained discriminant model

Classification is concerned with the development of rules for the allocation of observations to groups, and is a fundamental problem in machine learning. Much of previous work on classification models investigates two-group discrimination. Multi-category classification is less-often considered due to the tendency of generalizations of two-group models to produce misclassification rates that are higher than desirable. Indeed, producing “good” two-group classification rules is a challenging task for some applications, and producing good multi-category rules is generally more difficult. Additionally, even when the “optimal” classification rule is known, inter-group misclassification rates may be higher than tolerable for a given classification model. We investigate properties of a mixed-integer programming based multi-category classification model that allows for the pre-specification of limits on inter-group misclassification rates. The mechanism by which the limits are satisfied is the use of a reserved judgment region, an artificial category into which observations are placed whose attributes do not sufficiently indicate membership to any particular group. The method is shown to be a consistent estimator of a classification rule with misclassification limits, and performance on simulated and real-world data is demonstrated.     

Stability of the antiferromagnetic state of finite spin chains

The magnetic field strength at which the antiferromagnetic ordering becomes unstable is found analytically for a model of a one-dimensional ferromagnet with classical spin and anisotropic interaction. Such instabilities are inherent for the surface spin-flop transition, which is actively studied in hyperlattices. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115, No. 2, pp 275–279, May, 1998