Induction=I-Axiomatization+First-Order Consistency

In the early 1980s, there was a number of papers on what should be called proofs by consistency. They describe how to perform inductive proofs, without using an explicit induction scheme, in the context of equational specifications and ground-convergent rewrite systems. The method was explicitly stated as a first-order consistency proof in the case of pure equational, constructor-based specifications. In this paper, we show how, in general, inductive proofs can be reduced to first-order consistency and hence be performed by a first-order theorem prover. Moreover, we extend previous methods, allowing nonequational specifications (even non-Horn specifications) and designing some specific strategies. Finally, we also show how to drop the ground convergence requirement (which is called Saturatedness for general clauses).     

Fast cross-validation of high-breakdown resampling methods for PCA

Cross-validation (CV) is a very popular technique for model selection and model validation. The general procedure of leave-one-out CV (LOO-CV) is to exclude one observation from the data set, to construct the fit of the remaining observations and to evaluate that fit on the item that was left out. In classical procedures such as least-squares regression or kernel density estimation, easy formulas can be derived to compute this CV fit or the residuals of the removed observations. However, when high-breakdown resampling algorithms are used, it is no longer possible to derive such closed-form expressions. High-breakdown methods are developed to obtain estimates that can withstand the effects of outlying observations. Fast algorithms are presented for LOO-CV when using a high-breakdown method based on resampling, in the context of robust covariance estimation by means of the MCD estimator and robust principal component analysis. A robust PRESS curve is introduced as an exploratory tool to select the number of principal components. Simulation results and applications on real data show the accuracy and the gain in computation time of these fast CV algorithms.     

A rank-frequency model for scientific productivity

If f(r) is the number of contributions of an author or rank r, then it is shown that f(r) is proportional to r^–, where >0. The model is dependent on the definitions of a contribution and rank of an author. Three estimation procedures are illustrated and four other scientific productivity studies and two data sets on Canadian Mathematicians are shown to adequately fit this rank-frequency relationship.Research partially supported by National Research Council Grant No. A9229.     

Representations by Contact and Intersection of Segments

A necessary and sufficient condition is given for a connected bipartite graph to be the incidence graph of a contact family of segments and points. We deduce that any four-connected three-colorable plane graph is the contact graph of a family of segments and that any four-colored planar graph without an induced C₄ using four colors is the intersection graph of a family of straight line segments.