Spacing and shape of random peaks in non-parametric spectrum estimates

In this paper, expressions are derived for the expected number of spurious peaks in a spectrum estimate, that is, crossings above a given significance level per frequency unit, as well as the expected width of these peaks. In numerous scientific applications, spectrum estimates are used for the purpose of identifying sinusoidal or modal components, often thinning large sets of candidate frequencies with coincidence detection. Because one always expects numerous false peaks in a spectrum estimate, knowing the expected rate of false peaks helps to decide whether the number observed is abnormal and hence determine the true nature of the process. An example using solar wind data from the Advanced Composition Explorer is given where spectra display pathological numbers of significant peaks, while temporally permuted versions of the data possess spectra with the number expected for a white, Gaussian process. The permutation test is a valuable diagnostic for processes suspected to contain many line components.

Optimal designs for nonparametric estimation of zeros of regression functions

ABSTRACT We investigate nonparametric estimators of zeros of a regression function and its derivatives and we derive the distribution of design points minimizing the expected width of a confidence interval and the expected variance of the proposed estimator.

Profiles of random trees: correlation and width of random recursive trees and binary search trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

Profiles of random trees: correlation and width of random recursive trees and binary search trees

In a tree, a level consists of all those nodes that are the same distance from the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees. These coefficients undergo sharp sign-changes when one level is fixed and the other is varying. We also propose a new means of deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by singularity analysis.

Profiles of random trees: plane-oriented recursive trees

International audience We summarize several limit results for the profile of random plane-oriented recursive trees. These include the limit distribution of the normalized profile, asymptotic bimodality of the variance, asymptotic approximations of the expected width and the correlation coefficients of two level sizes. We also unveil an unexpected connection between the profile of plane-oriented recursive trees (with logarithmic height) and that of random binary trees (with height proportional to the square root of tree size).

Clinical Results after Percutaneous Automated Lumbar Nucleotomy

This study was carried out in order to assess the clinical results after percutaneous automated nucleotomy with regard to predictive factors for the outcome. Selection criteria included patients with predominance of low-back pain, diffuse posterior disk bulges and concomitant spinal stenosis who are not normally accepted for nucleotomy. In all, 142 patients were treated. The overall success rate after an average observation time of 21 months was 56%. The results were not influenced by whether a diffuse posterior bulge or a focal hernia had been treated or not (p=0.449). Spinal stenosis (p=0.043) and disk space narrowing exceeding 25% of the expected width (p=0.017) were associated with a poor outcome. By excluding these categories and patients with symptoms from more than one disk level, the success rate rose to 70%. With this selection, the results were equally good in patients with predominantly low-back pain compared to those with predominantly sciatica (p=0.490).

On the expected width function for topologically random channel networks

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.

On the expected width function for topologically random channel networks

An idealized river-channel network is represented by a trivalent planted plane tree, the root of which corresponds to the outlet of the network. A link of the network is any segment between a source and a junction, two successive junctions, or the outlet and a junction. For any x≧0, the width of the network is the number of links with the property that the distance of the downstream junction from the outlet is ≦x, and the distance of the upstream junction to the outlet is > x. Expressions are obtained for the expected width conditioned on N, (N, M), and (N, D), where N is the magnitude, M the order, and D the diameter of the network, under the assumption that the network is drawn from an infinite topologically random population and the link lengths are random.