A Data-Driven McMillan Degree Lower Bound

Given measurements of a linear time-invariant system, the McMillan degree is the dimension of the smallest such system that reproduces these observed dynamics. Using impulse response measurements where the system has been started in some (unknown) state and then allowed to evolve freely, a classical result by Ho and Kalman reveals the McMillan degree as the rank of a Hankel matrix built from these measurements. However, if measurements are contaminated by noise, this Hankel matrix will almost surely be full rank. Hence practitioners often estimate the rank of this matrix---and thus the McMillan degree---by manually setting a threshold between the large singular values that correspond to the non-zero singular values of the noise-free Hankel matrix and the small singular values that are pertubations of the zero singular values. Here we introduce a probabilistic upper bound on the perturbation of the singular values of this Hankel matrix when measurements are corrupted by additive Gaussian noise, and hence provide guidance on setting the threshold to obtain a lower bound on the McMillan degree. This result is powered by a new, probabilistic bound on the 2-norm of a random Hankel matrix with normally distributed entries. Unlike existing results for random Hankel matrices, this bound features no unknown constants and, moreover, is within a small factor of the empirically observed bound when entries are independent and identically distributed. This bound on the McMillan degree provides an inexpensive alternative to more general model order selection techniques such as the Akaike Information Criteria (AIC)

Excessive gas exchange impairment during exercise in a subject with a history of bronchopulmonary dysplasia and high altitude pulmonary edema

A 27-year-old male subject (V(O2 max)), 92% predicted) with a history of bronchopulmonary dysplasia (BPD) and a clinically documented case of high altitude pulmonary edema (HAPE) was examined at rest and during exercise. Pulmonary function testing revealed a normal forced vital capacity (FVC, 98.1% predicted) and diffusion capacity for carbon monoxide (D(L(CO)), 91.2% predicted), but significant airway obstruction at rest [forced expiratory volume in 1 sec (FEV(1)), 66.5% predicted; forced expiratory flow at 50% of vital capacity (FEF(50)), 34.3% predicted; and FEV(1) /FVC 56.5%] that was not reversible with an inhaled bronchodilator. Gas exchange worsened from rest to exercise, with the alveolar to arterial P(O2) difference (AaD(O2)) increasing from 0 at rest to 41 mmHg at maximal normoxic exercise (VO(2) = 41.4 mL/kg/min) and from 11 to 31 mmHg at maximal hypoxic exercise (VO(2) = 21.9 mL/kg/min). Arterial P(O2) decreased to 67.8 and 29.9 mmHg at maximal normoxic and hypoxic exercise, respectively. These data indicate that our subject with a history of BPD is prone to a greater degree of exercise-induced arterial hypoxemia for a given VO(2) and F(I(O2)) than healthy age-matched controls, which may increase the subject's susceptibility to high altitude illness

An alternative hypothesis testing strategy for secondary phenotype data in case-control genetic association studies

Motivated by the challenges associated with accounting for the ascertainment when analyzing secondary phenotypes that are correlated with case-control status, Lin and Zeng have proposed a method that properly reflects the case-control sampling (Lin and Zeng, 2009). The Lin and Zeng method has the advantage of accurately estimating effect sizes for secondary phenotypes that are normally distributed or dichotomous. This method can be computationally intensive in practice under the null hypothesis when the likelihood surface that needs to be maximized can be relatively flat. We propose an extension of the Lin and Zeng method for hypothesis testing that uses proportional odds logistic regression to circumvent these computational issues. Through simulation studies, we compare the power and type-1 error rate of our method to standard approaches and Lin and Zeng's approach

EXPLORING THE RELATION BETWEEN HIGH CREATIVITY AND HIGH ACHIEVEMENT AMONG 8TH AND 11TH GRADERS

This study explores the relationship between high creativity and high scholastic achievement in mathematics, reading, and science among 8th and 11th grade students. A quantitative methodology was used in the study. Data were collected from 941 eighth-grade students and 605 eleventh-grade students at an independent public school district in Minnesota, a Midwestern State in the U.S.A. The data collection instruments included a general personal questionnaire, the Torrance Tests of Creative Thinking (TTCT) Figural Form A, and the Minnesota Comprehensive Assessment Test (MCA). Correlations were computed and chi-square analyses were performed to address research questions. Grade-based standard scores were used in the measurement of creativity and academic achievement with the top 20% cutoff scores being used to identify students with high achievement and students with high creativity. The results indicated that the relation between high creativity and high academic achievement varies among eighth and eleventh graders. High mathematics and high reading achievement are related to high creativity among both eighth- and eleventh-grade students, but with small effect sizes. High achievement in science is related to high creativity among eleventh-graders. Our results indicate that high creativity and high achievement in reading, mathematics, and science achievement tend to be positively related, but those relationships are at best weak, indicating that there are substantial components to high creativity that are not shared by high achievement in mathematics, reading, and science and vice versa. Keywords: creativity, academic achievement, eighth grade, eleventh grade &nbsp

Magnetic damping of an elastic conductor

Many applications call for a design that maximizes the rate of energy decay. Typical problems of this class include one dimensional damped wave operators, where energy dissipation is caused by a damping operator acting on the velocity. Two damping operators are well understood: a multiplication operator (known as viscous damping) and a scaled Laplacian (known as Kelvin---Voigt damping). Paralleling the analysis of viscous damping, this thesis investigates energy decay for a novel third operator known as magnetic damping, where the damping is expressed via a rank-one self-adjoint operator, dependent on a function a. This operator describes a conductive monochord embedded in a spatially varying magnetic field perpendicular to the monochord and proportional to a. Through an analysis of the spectrum, this thesis suggests that unless a has a singularity at one boundary for any finite time, there exist initial conditions that give arbitrarily small energy decay at any time

A general semi-parametric approach to the analysis of genetic association studies in population-based designs

Background: For genetic association studies in designs of unrelated individuals, current statistical methodology typically models the phenotype of interest as a function of the genotype and assumes a known statistical model for the phenotype. In the analysis of complex phenotypes, especially in the presence of ascertainment conditions, the specification of such model assumptions is not straight-forward and is error-prone, potentially causing misleading results. Results: In this paper, we propose an alternative approach that treats the genotype as the random variable and conditions upon the phenotype. Thereby, the validity of the approach does not depend on the correctness of assumptions about the phenotypic model. Misspecification of the phenotypic model may lead to reduced statistical power. Theoretical derivations and simulation studies demonstrate both the validity and the advantages of the approach over existing methodology. In the COPDGene study (a GWAS for Chronic Obstructive Pulmonary Disease (COPD)), we apply the approach to a secondary, quantitative phenotype, the Fagerstrom nicotine dependence score, that is correlated with COPD affection status. The software package that implements this method is available. Conclusions: The flexibility of this approach enables the straight-forward application to quantitative phenotypes and binary traits in ascertained and unascertained samples. In addition to its robustness features, our method provides the platform for the construction of complex statistical models for longitudinal data, multivariate data, multi-marker tests, rare-variant analysis, and others