On approximating optimal weight “no”-certificates in weighted difference constraint systems

This paper is concerned with the design and analysis of approximation algorithms for the problem of determining the least weight refutation in a weighted difference constraint system. Recall that a difference constraint is a linear constraint of the form \(x_{i}-x_{j} \le b_{ij}\) and a conjunction of such constraints is called a difference constraint system (DCS). In a weighted DCS (WDCS), a positive weight is associated with each constraint. Every infeasible constraint system has a refutation, which attests to its infeasibility. In the case of a DCS, this refutation is a subset of the input constraints, which when added together produces a contradiction of the form \(0 \le -b\), \(b> 0\). It follows that every refutation acts as a “no”-certificate. The length of a refutation is the number of constraints used in the derivation of a contradiction. Associated with a DCS \(\mathbf{D: A\cdot x \le b}\) is its constraint network \(\mathbf{G= \langle V,E, b \rangle }\). It is well-known that \(\mathbf{D}\) is infeasible if and only if \(\mathbf{G}\) contains a simple, negative cost cycle. Previous research has established that every negative cost cycle of length k in \(\mathbf{G}\) corresponds exactly to a refutation of \(\mathbf{D}\) using k constraints. It follows that the shortest refutation of \(\mathbf{D}\) (i.e., the refutation which uses the fewest number of constraints) corresponds to the length of the shortest negative cycle in \(\mathbf{G}\). The constraint network of a WDCS is represented by a constraint network \(\mathbf{G = \langle V, E, b, l \rangle }\), where \(\mathbf{l}:\mathbf{E \rightarrow \mathbb {N}}\) represents a function which associates a positive, integral length with each edge in \(\mathbf{G}\). In the case of a WDCS, the weight of a refutation is defined as the sum of the lengths of the edges corresponding to the refutation. The problem of finding the minimum weight refutation in a WDCS is called the weighted optimal length resolution refutation (WOLRR) problem and is known to be NP-hard. In this paper, we describe a pseudo-polynomial time algorithm for the WOLRR problem and convert it into a fully polynomial time approximation scheme (FPTAS).     

A bayesian approach for testing periodicity of dichotomous observations

A Bayesian approach is utilized to test for periodicity in a dichotomous time series. Dichotomous data arise in a variety of circumstances when a variable takes on only two possible values. Conjugate and noninformative priors are considered as well as a hierarchical Bayes approach; the latter is considered the superior Bayes methodology. The situation of stochastic period lengths is also discussed. The generalization to the multinomial model is investigated to allow for the case that a variable takes on more than two possible values. In all cases decisions are made based on a Bayes factor. The proposed procedures are demonstrated on earthquake data in the central Virginia seismic zone     

Estimating detailed distributions from grouped sociodemographic data: ‘get me started in’ curve fitting using nonlinear models

In much demographic analysis, it is important to know how occurrence-exposure rates or transition probabilities vary continuously by age or by time. Often we have coarse or fluctuating data so there can be a need for estimation and smoothing. Since the distributions of rates or counts across age or another variable are often curved, a nonlinear model is likely to be appropriate. The main focus of this paper is on the estimation of detailed information from grouped data such as age and income bands; however, the methods we outline could also be applied to other settings such as smoothing rates where the original data are ragged. The ability to carry out curve fitting is a very useful skill for population geographers and demographers. Curve fitting is not well covered in statistics textbooks, and whilst there is a large literature in journals thoroughly discussing the detail of functions which define curves, these texts are likely to be inaccessible to researchers who are not specialists in mathematics. We aim here to make nonlinear modelling as accessible as possible. We demonstrate how to carry out nonlinear regression using SPSS, giving stepped-through hypothetical and research examples. We note other software in which nonlinear regression can be carried out, and outline alternative methods of curve fitting.     

On the possible significance of the quadratic effect of age on lung-function measurements

The problem of interpreting lung-function measurements in industrial workers is examined. Two common lung-function measurements (FEV₁, and FVC) are described. The standard method currently used in the analysis of such cross-sectional survey data is discussed. The basic assumption of a linear decline with age is questioned on the basis of large sets of data from a variety of industries in British Columbia. It is shown that, while the linear assumption holds approximately in unexposed. healthy nonsmoking individuals, a quadratic age effect is often observed in smokers and/or in individuals who are industrially exposed to certain fumes or dusts. Recognizing this accelerated rate of deterioration in the lungs is of fundamental importance both to the identification of affected individuals and to the understanding of the process involved. An attempt is made to interpret the variety of nonlinear situations observed, by appealing to population selection mechanisms, individual variations in susceptibility, and the effects due to various levels of stimulus strength.     

Estimation of the Population Size in the Classical Occupancy Problem

In the classical occupancy problem where the random variable X is the number of N elements selected by K individuals when each element is equally likely to be chosen by any of the individuals, it is desired to estimate N. Three estimators given in the literature are compared with three estimators derived in this article, two of which are based on Bayesian methods, utilizing a simulation study. One of the Bayes estimators appears to perform the best along with one proposed in the literature in 1968. The estimators are also compared using data obtained from a cemetery in Ohio.     

Non-estrogen-based treatments for menopausal symptoms

The British Menopause Society Council is committed to provide up-to-date authoritative reviews to aid health professionals to inform and advise women about key issues in postreproductive health. This guidance refers to non-estrogen-based treatments for menopausal symptoms, such as hot flushes, symptoms of urogenital atrophy and lack of sexual desire. Treatment of choice should be based on up to-date information and targeted to individual women's needs. Non-hormonal strategies may be useful for women with estrogen-dependent disease such as breast cancer.     

Developing strategies for deriving small population fertility rates

Many agencies require population estimates and projections by ethnic group. These projections need ethnic-specific, age-specific fertility rates (ASFRs) but their inclusion is challenging since ethnicity is not recorded at birth registration. In this paper maternity data are used in a case study of electoral wards in Bradford, West Yorkshire, to develop fertility rates for small populations for a 1991 based projection. The challenge is to capture local variations in fertility by ethnic group when data are sparse. Small areas were grouped together using cluster analysis to define combinations with similar sociodemographic and fertility experiences so that sparse data could be aggregated to estimate reliable ethnic-specific fertility rates. For comparison, the data were aggregated into the 1991 Office for National Statistics area type classification. Fertility rates by single year of age for all area types were smoothed using the Hadwiger function. For the White ethnic group there were sufficient births to create ethnic-specific, ward-level ASFRs. For other ethnicities grouping of areas was necessary. The accuracy of the ASFRs in predicting births was assessed using mean absolute percentage error. Results show that for some minority groups district-level ethnic-specific fertility rates produced the most accurate birth estimates even though they were based on a larger area. This implies that rates created may be informative about the local area for White ethnic type but not in the same way for smaller ethnic groups. In terms of grouping strategies we recommend that existing classifications are assessed to determine how well variations in rates are stratified before embarking on a custom scheme. Where population sub-groups are small in some areas, it may be more reliable to use rates derived for larger areas and apply these to local populations. Inevitably, the rates used in a projection are a compromise but hopefully will still capture important dimensions of population change.