Some results concerning off-training-set and IID error for the Gibbs and the Bayes optimal generalizers

In this paper we analyse the average behaviour of the Bayes-optimal and Gibbs learning algorithms. We do this both for off-training-set error and conventional IID (independent identically distributed) error (for which test sets overlap with training sets). For the IID case we provide a major extension to one of the better known results. We also show that expected IID test set error is a non-increasing function of training set size for either algorithm. On the other hand, as we show, the expected off-training-set error for both learning algorithms can increase with training set size, for non-uniform sampling distributions. We characterize the relationship the sampling distribution must have with the prior for such an increase. We show in particular that for uniform sampling distributions and either algorithm, the expected off-training-set error is a non-increasing function of training set size. For uniform sampling distributions, we also characterize the priors for which the expected error of the Bayes-optimal algorithm stays constant. In addition we show that for the Bayes-optimal algorithm, expected off-training-set error can increase with training set size when the target function is fixed, but if and only if the expected error averaged over all targets decreases with training set size. Our results hold for arbitrary noise and arbitrary loss functions.     

Individual Growth Curves of Frequency of Sexual Intercourse Among Urban,Adolescent, African American Youth

Summary

In the current study we examined individual growth curves of frequency of sexual intercourse among a sample of urban, low-income, African American youth at increased risk for subsequent HIV/AIDS exposure. Three waves of longitudinal data from the Collaborative HIV-Prevention AdolescentMental Health (CHAMP) project were utilized. Participant ages ranged from 9 to 12 years (M = 11 years) at the first interview wave and from 15 to 19 years (M = 18 years) at the final interview wave. As such, we were able to map out true developmental trajectories of sexual intercourse over a 10-year period of adolescence (spanning ages 9 to 19 years). Results indicate that the average study participant was sexually abstinent (in terms of intercourse) during the pre-teen years, reported a single episode of sexual intercourse between ages 14 and 15, and by age 19, reported between 3 and 10 episodes of sexual intercourse. Significant variability in the acceleration of growth rates (as captured by a quadratic random effect) was observed, suggesting that some youth accelerated more rapidly (in their sexual intercourse histories) than did others. Participant gender predicted trajectory starting points; boys reported higher rates of sexual intercourse at age 12. Frequency of baseline exposure to sexual possibility situations (i.e., being in mixed-sex company in a private place in the absence of adult supervision) predicted growth curve acceleration, suggesting preteenswith more exposure to sexual possibility situations accelerated more rapidly in their rates of sexual intercourse over time. Developmental implications of these data are discussed.     

Replication of Low-Risk Gambling Limits Using Canadian Provincial Gambling Prevalence Data

A set of low-risk gambling limits were recently produced using Canadian epidemiological data on the intensity of gambling behavior and related consequences (Currie et al. Addiction 101:570–580, 2006). The empirically derived limits (gambling no more than two to three times per month, spending no more than $501–$100°CAN per year or no more than 1% of gross income spent on gambling) accurately predicted risk of gambling-related harm after controlling for other risk factors. The present study sought to replicate these limits on data collected in three independently conducted Canadian provincial gambling surveys. Dose–response curves and logistic regression analyses were applied to gambling prevalence data collected in surveys conducted in 2001–2002 within the provinces of Alberta, British Columbia, and Ontario (combined sample N = 7,675). A comparable dose–response relationship between gambling intensity and risk of harm was found in each province. The optimal thresholds for defining an upper limit of low-risk gambling were similar across the three provinces despite variations in the availability and organization of legalized gambling opportunities within each region. These results provide additional evidence supporting the validity of the low-risk gambling limits. Quantitative limits could be used to augment existing responsible gambling guidelines.

Principal curves revisited

A principal curve (Hastie and Stuetzle, 1989) is a smooth curve passing through the middle of a distribution or data cloud, and is a generalization of linear principal components. We give an alternative definition of a principal curve, based on a mixture model. Estimation is carried out through an EM algorithm. Some comparisons are made to the Hastie-Stuetzle definition.     

On the confusion matrix in credit scoring and its analytical properties

Abstract

Confusion Matrix is an important measure to evaluate the accuracy of credit scoring models. However, the literature about Confusion Matrix is limited. The analytical properties of Confusion Matrix are ignored. Moreover, the concept of Confusion Matrix is confusing. In this article, we systematically study Confusion Matrix and its analytical properties. We enumerate 16 possible variants of Confusion Matrix and show that only 8 are reasonable. We study the relationship between Confusion Matrix and 2 other performance measures: the receiver operating characteristic curve (ROC) and Kolmogorov-Smirnov statistic (KS). We show that an optimal cutoff score can be attained by KS.     

A Framework for Analysis of Production Authorization Card‐Controlled Production Systems

A framework for the analysis of manufacturing systems operating under a production authorization card (PAC) system is outlined. The PAC system provides a single model, which encompasses a broad variety of control strategies, including Kanban and CONWIP. This paper describes a framework for the performance analysis and comparison of both specific and families of control strategies. The framework starts with system performance measures estimated by simulation. These simulations in turn provide training data for neural network metamodels. The metamodels allow for a variety of analysis and optimization approaches, including the construction of optimal policy curves, which can provide considerable insight into the systems under study.     

The cost of not knowing the radius

Robust Statistics considers the quality of statistical decisions in the presence of deviations from the ideal model, where deviations are modelled by neighborhoods of a certain size about the ideal model. We introduce a new concept of optimality (radius-minimaxity) if this size or radius is not precisely known: for this notion, we determine the increase of the maximum risk over the minimax risk in the case that the optimally robust estimator for the false neighborhood radius is used. The maximum increase of the relative risk is minimized in the case that the radius is known only to belong to some interval [r _l ,r _u ]. We pursue this minmax approach for a number of ideal models and a variety of neighborhoods. Also, the effect of increasing parameter dimension is studied for these models. The minimax increase of relative risk in case the radius is completely unknown, compared with that of the most robust procedure, is 18.1% versus 57.1% and 50.5% versus 172.1% for one-dimensional location and scale, respectively, and less than 1/3 in other typical contamination models. In most models considered so far, the radius needs to be specified only up to a factor , in order to keep the increase of relative risk below 12.5%, provided that the radius–minimax robust estimator is employed. The least favorable radii leading to the radius–minimax estimators turn out small: 5–6% contamination, at sample size 100.      

Testing equality of survival distributions when the population marks are missing

This paper introduces a nonparametric approach for testing the equality of two or more survival distributions based on right censored failure times with missing population marks for the censored observations. The standard log-rank test is not applicable here because the population membership information is not available for the right censored individuals. We propose to use the imputed population marks for the censored observations leading to fractional at-risk sets that can be used in a two sample censored data log-rank test. We demonstrate with a simple example that there could be a gain in power by imputing population marks (the proposed method) for the right censored individuals compared to simply removing them (which also would maintain the right size). Performance of the imputed log-rank tests obtained this way is studied through simulation. We also obtain an asymptotic linear representation of our test statistic. Our testing methodology is illustrated using a real data set.     

Shape analysis for automated sulcal classification and parcellation of MRI data

We describe geometric invariants that characterize the shape of curves and surfaces in 3D space: curvature, Gauss integrals and moments. We apply these invariants to neuroimaging data to determine if they have application for automatically classifying and parcellating cortical data. The curves of sulci and gyri on the cortical surface can be obtained by reconstructing cortical surface representations of the human brain from magnetic resonance imaging (MRI) data. We reconstructed gray matter surfaces for 15 subjects, traced 10 sulcal curves on each surface and computed geometric invariants for each curve. These geometric features were used classify the curves into sulcal and hemispheric classes. The best classification results were obtained when moment-based features were computed on the sulcal curves in native space. Gauss integral measures showed that they were useful for differentiating the hemispheric location of a single sulcus. These promising results may indicate that moment invariants are useful for characterizing shape on a global scale. Gauss integral invariants are potentially useful measures for characterizing cortical shape on a local, rather than global scale. Gauss integrals have found biological significance in characterizing proteins so it is worthwhile to consider their possible application to neuroscientific data.