Generating All Permutations by Context-Free Grammars in Greibach Normal Form

We consider context-free grammars $G_n$ in Greibach normal form and, particularly, in Greibach $m$-form ($m=1,2$) which generates the finite language $L_n$ of all $n!$ strings that are permutations of $n$ different symbols ($n\geq 1$). These grammars are investigated with respect to their descriptional complexity, i.e., we determine the number of nonterminal symbols and the number of production rules of $G_n$ as functions of $n$. As in the case of Chomsky normal form these descriptional complexity measures grow faster than any polynomial function

Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests

The estimation of variance-based importance measures (called Sobol' indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the Derivative-based Global Sensitivity Measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol' indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice

Predicting the required number of training samples

A criterion which measures the quality of the estimate of the covariance matrix of a multivariate normal distribution is developed. Based on this criterion, the necessary number of training samples is predicted. Experimental results which are used as a guide for determining the number of training samples are included

Gaussian limits for multidimensional random sequential packing at saturation (extended version)

Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume $\lambda$ is asymptotically normal as $\lambda \to \infty$. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization.Comment: 31 page

What accuracy statistics really measure

Provides the software estimation research community with a better understanding of the meaning of, and relationship between, two statistics that are often used to assess the accuracy of predictive models: the mean magnitude relative error (MMRE) and the number of predictions within 25% of the actual, pred(25). It is demonstrated that MMRE and pred(25) are, respectively, measures of the spread and the kurtosis of the variable z, where z=estimate/actual. Thus, z is considered to be a measure of accuracy, and statistics such as MMRE and pred(25) to be measures of properties of the distribution of z. It is suggested that measures of the central location and skewness of z, as well as measures of spread and kurtosis, are necessary. Furthermore, since the distribution of z is non-normal, non-parametric measures of these properties may be needed. For this reason, box-plots of z are useful alternatives to simple summary metrics. It is also noted that the simple residuals are better behaved than the z variable, and could also be used as the basis for comparing prediction system