Stochastic Data Clustering

In 1961 Herbert Simon and Albert Ando published the theory behind the long-term behavior of a dynamical system that can be described by a nearly uncoupled matrix. Over the past fifty years this theory has been used in a variety of contexts, including queueing theory, brain organization, and ecology. In all these applications, the structure of the system is known and the point of interest is the various stages the system passes through on its way to some long-term equilibrium. This paper looks at this problem from the other direction. That is, we develop a technique for using the evolution of the system to tell us about its initial structure, and we use this technique to develop a new algorithm for data clustering.Comment: 23 page

Chuck Wessell, Assistant Professor of Mathematics

In this newest Next Page column, Chuck Wessell, Assistant Professor of Mathematics, shares what he and his cat, George, read first thing in the morning; his affinity for books with colons in the title; must-read math books for the non-mathematician; and much more

Reducing the Effects of Unequal Number of Games on Rankings

Ranking is an important mathematical process in a variety of contexts such as information retrieval, sports and business. Sports ranking methods can be applied both in and beyond the context of athletics. In both settings, once the concept of a game has been defined, teams (or individuals) accumulate wins, losses, and ties, which are then factored into the ranking computation. Many settings involve an unequal number of games between competitors. This paper demonstrates how to adapt two sports rankings methods, the Colley and Massey ranking methods, to settings where an unequal number of games are played between the teams. In such settings, the standard derivations of the methods can produce nonsensical rankings. This paper introduces the idea of including a super-user into the rankings and considers the effect of this fictitious player on the ratings. We apply such techniques to rank batters and pitchers in Major League baseball, professional tennis players, and participants in a free online social game. The ideas introduced in this paper can further the scope that such methods are applied and the depth of insight they offer

270: How to Win the Presidency with Just 17.56% of the Popular Vote

With the U.S. presidential election fast approaching we will often be reminded that the candidate who receives the most votes is not necessarily elected president. Instead, the winning candidate must receive a majority of the 538 electoral votes awarded by the 50 states and the District of Columbia. Someone with a curious mathematical mind might then wonder: What is the small fraction of the popular vote a candidate can receive and still be elected president? [excerpt

Diagnostic performance and reliability of ultrasonography for fatty degeneration of the rotator cuff muscles

BACKGROUND: Diagnostic evaluation of rotator cuff muscle quality is important to determine indications for potential operative repair. Ultrasonography has developed into an accepted and useful tool for evaluating rotator cuff tendon tears; however, its use for evaluating rotator muscle quality has not been well established. The purpose of this study was to investigate the diagnostic performance and observer reliability of ultrasonography in grading fatty degeneration of the posterior and superior rotator cuff muscles. METHODS: The supraspinatus, infraspinatus, and teres minor muscles were prospectively evaluated with magnetic resonance imaging (MRI) and ultrasonography in eighty patients with shoulder pain. The degree of fatty degeneration on MRI was graded by four independent raters on the basis of the modified Goutallier grading system. Ultrasonographic evaluation of fatty degeneration was performed by one of three radiologists with use of a three-point scale. The two scoring systems were compared to determine the diagnostic performance of ultrasonography. The interobserver and intraobserver reliability of MRI grading by the four raters were determined. The interobserver reliability of ultrasonography among the three radiologists was determined in a separate group of thirty study subjects. The weighted Cohen kappa, percentage agreement, sensitivity, and specificity were calculated. RESULTS: The accuracy of ultrasonography for the detection of fatty degeneration, as assessed on the basis of the percentage agreement with MRI, was 92.5% for the supraspinatus and infraspinatus muscles and 87.5% for the teres minor. The sensitivity was 84.6% for the supraspinatus, 95.6% for the infraspinatus, and 87.5% for the teres minor. The specificity was 96.3% for the supraspinatus, 91.2% for the infraspinatus, and 87.5% for the teres minor. The agreement between MRI and ultrasonography was substantial for the supraspinatus and infraspinatus (kappa = 0.78 and 0.71, respectively) and moderate for the teres minor (kappa = 0.47). The interobserver reliability for MRI was substantial for the supraspinatus and infraspinatus (kappa = 0.76 and 0.77, respectively) and moderate for the teres minor (kappa = 0.59). For ultrasonography, the interobserver reliability was substantial for all three muscles (kappa = 0.71 for the supraspinatus, 0.65 for the infraspinatus, and 0.72 for the teres minor). CONCLUSIONS: The diagnostic performance of ultrasonography in identifying and grading fatty degeneration of the rotator cuff muscles was comparable with that of MRI. Ultrasonography can be used as the primary diagnostic imaging modality for fatty changes in rotator cuff muscles. LEVEL OF EVIDENCE: Diagnostic Level II. See Instructions for Authors for a complete description of levels of evidence

A Nonnegative Analysis of Politics

The article investigates how linear algebra can recover mathematical information from the electronic messages using the Enron Email Sets in Pennsylvania. It states that a term-by-email matrix has been created to cluster algorithm, which allows one to mine through data and discover meaning. Moreover, nonnegative matrix factorization (NMF) enables one to interpret the resulting factorization in terms of the original problem

ICAR: endoscopic skull‐base surgery

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Habitat Assessment of Non-Wadeable Rivers in Michigan

Habitat evaluation of wadeable streams based on accepted protocols provides a rapid and widely used adjunct to biological assessment. However, little effort has been devoted to habitat evaluation in non-wadeable rivers, where it is likely that protocols will differ and field logistics will be more challenging. We developed and tested a non-wadeable habitat index (NWHI) for rivers of Michigan, where non-wadeable rivers were defined as those of order ≥5, drainage area ≥1600 km 2 , mainstem lengths ≥100 km, and mean annual discharge ≥15 m 3 /s. This identified 22 candidate rivers that ranged in length from 103 to 825 km and in drainage area from 1620 to 16,860 km 2 . We measured 171 individual habitat variables over 2-km reaches at 35 locations on 14 rivers during 2000–2002, where mean wetted width was found to range from 32 to 185 m and mean thalweg depth from 0.8 to 8.3 m. We used correlation and principal components analysis to reduce the number of variables, and examined the spatial pattern of retained variables to exclude any that appeared to reflect spatial location rather than reach condition, resulting in 12 variables to be considered in the habitat index. The proposed NWHI included seven variables: riparian width, large woody debris, aquatic vegetation, bottom deposition, bank stability, thalweg substrate, and off-channel habitat. These variables were included because of their statistical association with independently derived measures of human disturbance in the riparian zone and the catchment, and because they are considered important in other habitat protocols or to the ecology of large rivers. Five variables were excluded because they were primarily related to river size rather than anthropogenic disturbance. This index correlated strongly with indices of disturbance based on the riparian (adjusted R 2 = 0.62) and the catchment (adjusted R 2 = 0.50), and distinguished the 35 river reaches into the categories of poor (2), fair (19), good (13), and excellent (1). Habitat variables retained in the NWHI differ from several used in wadeable streams, and place greater emphasis on known characteristic features of larger rivers.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41269/1/267_2004_Article_141.pd