Separation and probability of correct classification among two or more distributions

A function on theK-fold product of a set in normed vector space will be called a separation measurement if, for any collection ofK points, the function is bounded below and above, respectively, by maximum and total distance between pairs of points in the collection. Separation measurements are relavent toK-sample hypothesis testing and also to discrimination amongK classes, and several examples are given. In particular, ordinaryL ₁ distance between integrable functions can be generalized to a non-pairwise separation measurement for densitiesf ₁,f ₂,...,f _K inL ₁[μ]; and this separation is a linear transform of the optimal discriminant's probability of correct classification. This research was supported by grant A8044 from the National Research Council of Canada.     

On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the ??yea?? voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243?C260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9?voters which do not admit a unique minimum sum integer weight representation.     

Monotonicity of power in weighted voting games with restricted communication

Indices that evaluate the distribution of power in simple games are commonly required to be monotonic in voting weights when the game represents a voting body such as a shareholder meeting, parliament, etc. The standard notions of local or global monotonicity are bound to be violated, however, if cooperation is restricted to coalitions that are connected by a communication graph. This paper proposes new monotonicity concepts for power in games with communication structure and investigates the monotonicity properties of the Myerson value, the restricted Banzhaf value, the position value, and the average tree solution.     

A classification of the main probability distributions by minimizing the weighted logarithmic measure of deviation

The paper reanalyzes the following nonlinear program: Find the most similar probability distribution to a given reference measure subject to constraints expressed by mean values by minimizing the weighted logarithmic deviation. The main probability distributions are examined from this point of view and the results are summarized in a table.The author acknowledges the NSERC Canada Research Grant A-5712.     

Reliability analysis and optimization of weighted voting systems with continuous states input

Weighted voting systems are widely used in many practical fields such as target detection, human organization, pattern recognition, etc. In this paper, a new model for weighted voting systems with continuous state inputs is formulated. We derive the analytical expression for the reliability of the entire system under certain distribution assumptions. A more general Monte Carlo algorithm is also given to numerically analyze the model and evaluate the reliability. This paper further proposes a reliability optimization problem of weighted voting systems under cost constraints. A genetic algorithm is introduced and applied as the optimization technique for the model formulated. A numerical example is then presented to illustrate the ideas.     

On redundant weighted voting systems with components having stochastic arrangement increasing lifetimes

As one generalization of the k-out-of-n structure, the weighted voting system has been paid much attention during the past two decades. This paper has a further study on active redundancies allocation to weighted voting reliability systems of components having LWSAI lifetimes. For redundancies with SAI lifetimes, allocating a more reliable redundancy to a weaker and more heavily weighted component is found to produce a more reliable system in the sense of having higher reliability. Also, in the context of redundancies with identically distributed lifetimes, we show that allocating more redundancies to a weaker and more heavily weighted component produces a more reliable system. Some numerical examples are presented to illustrate the main results as well.     

Convergence rates for weighted sums in noncommutative probability space

We study convergence rates for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. As applications, we first study convergence rates for weighted sums of random variables in the noncommutative Lorentz space, and second we study convergence rates for weighted sums of probability measures with respect to the free additive convolution.     

Decision functions for chain classifiers based on Bayesian networks for multi-label classification

Multi-label classification problems require each instance to be assigned a subset of a defined set of labels. This problem is equivalent to finding a multi-valued decision function that predicts a vector of binary classes. In this paper we study the decision boundaries of two widely used approaches for building multi-label classifiers, when Bayesian network-augmented naive Bayes classifiers are used as base models: Binary relevance method and chain classifiers. In particular extending previous single-label results to multi-label chain classifiers, we find polynomial expressions for the multi-valued decision functions associated with these methods. We prove upper boundings on the expressive power of both methods and we prove that chain classifiers provide a more expressive model than the binary relevance method.     

Mathematical programming based heuristics for improving LP-generated classifiers for the multiclass supervised classification problem

Mathematical programming is used as a nonparametric approach to supervised classification. However, mathematical programming formulations that minimize the number of misclassifications on the design dataset suffer from computational difficulties. We present mathematical programming based heuristics for finding classifiers with a small number of misclassifications on the design dataset with multiple classes. The basic idea is to improve an LP-generated classifier with respect to the number of misclassifications on the design dataset. The heuristics are evaluated computationally on both simulated and real world datasets.     

Topological classification of spaces of probability measures for co-analytic sets

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 10–19, January, 1994.     

A simple “market value” bargaining model for weighted voting games: characterization and limit theorems

We offer a bargaining model for weighted voting games that is a close relative of the nucleolus and the kernel. We look for a set of weights that preserves winning coalitions that has the property of minimizing the difference between the weight of the smallest and the weight of the largest Minimum Winning Coalition. We claim that such a set of weights provides an a priori measure of a weighted voter’s bribeworthiness or market value. After introducing our model, we provide a characterization result for this model and show its links to other bargaining model approaches in the literature. Then we offer some limit results showing that, with certain reasonable conditions on the distributions of weights, as the size of the voting body increases, the values of bribeworthiness we calculate will approach both the weights themselves and the Banzhaf scores for the weighted voting game. We also show that, even for relatively small groups using weighted voting, such as the membership of the European Council of Ministers (and its predecessors) 1958–2003, similarities among the usual a priori power scores, bribeworthiness/market value, and the weights themselves, will be quite strong.     

Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang[7].     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper deals with the approximation of the tail probability of randomly weighted sums of a sequence of pairwise quasi-asymptotically independent but non-identically distributed dominatedly-varying-tailed random variables. The weights are independent of the former sequence, satisfying some assumptions about the moments. But no requirements on the dependence structure of the weights are imposed.     

A general model for voting systems with multiple alternatives

In this paper, the extension of simple games to the vector case is proposed. Games with multiple qualitative criteria and multi-criteria simple games are introduced as a natural tool for modelling voting systems and related social-choice situations. After formally defining these games, the special class of monotonic multi-criteria simple games is characterized. We show that these games enable the formulation and analysis of several collective decision models proposed in the literature. Furthermore, our model can be applied to group-decision problems which cannot be analyzed in the existing frameworks.     

Erratum to: Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

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