Modern Statistical Models and Methods for Estimating Fatigue-Life and Fatigue-Strength Distributions from Experimental Data

Engineers and scientists have been collecting and analyzing fatigue data since the 1800s to ensure the reliability of life-critical structures. Applications include (but are not limited to) bridges, building structures, aircraft and spacecraft components, ships, ground-based vehicles, and medical devices. Engineers need to estimate S-N relationships (Stress or Strain versus Number of cycles to failure), typically with a focus on estimating small quantiles of the fatigue-life distribution. Estimates from this kind of model are used as input to models (e.g., cumulative damage models) that predict failure-time distributions under varying stress patterns. Also, design engineers need to estimate lower-tail quantiles of the closely related fatigue-strength distribution. The history of applying incorrect statistical methods is nearly as long and such practices continue to the present. Examples include treating the applied stress (or strain) as the response and the number of cycles to failure as the explanatory variable in regression analyses (because of the need to estimate strength distributions) and ignoring or otherwise mishandling censored observations (known as runouts in the fatigue literature). The first part of the paper reviews the traditional modeling approach where a fatigue-life model is specified. We then show how this specification induces a corresponding fatigue-strength model. The second part of the paper presents a novel alternative modeling approach where a fatigue-strength model is specified and a corresponding fatigue-life model is induced. We explain and illustrate the important advantages of this new modeling approach.Comment: 93 pages, 27 page

On the Statistical Modeling and Analysis of Repairable Systems

We review basic modeling approaches for failure and maintenance data from repairable systems. In particular we consider imperfect repair models, defined in terms of virtual age processes, and the trend-renewal process which extends the nonhomogeneous Poisson process and the renewal process. In the case where several systems of the same kind are observed, we show how observed covariates and unobserved heterogeneity can be included in the models. We also consider various approaches to trend testing. Modern reliability data bases usually contain information on the type of failure, the type of maintenance and so forth in addition to the failure times themselves. Basing our work on recent literature we present a framework where the observed events are modeled as marked point processes, with marks labeling the types of events. Throughout the paper the emphasis is more on modeling than on statistical inference.Comment: Published at http://dx.doi.org/10.1214/088342306000000448 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Travelling waves in hyperbolic chemotaxis equations

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically

Uncertainty in the modeling of spatial big data on a pattern of bushfires holes

This paper focuses on the presence of vegetation patches, called holes remaining after forest fires. Holes are of interest to explore because their vegetation is affected by severe temperature stress nearby, although they can serve as an agent to regenerate a forest after the burn. Further, it is interesting to know why holes emerge at all, while little if anything is known about their structure and distribution in space. A statistical analysis of their presence and abundance and a spatial statistical analysis to analyze their positions was done within four forest fire footprints. Fractal dimension of the holes was compared to that of the forest fire footprint, whereas remote sensing imagery was used to identify the normalized difference vegetation index (NDVI) of the patches before and after the fire. Results showed that the fractal dimension of the holes is lower than that of the forest fire footprint, and that the NDVI is slowly recovering to the original NDVI. Differences with the NDVI of the surrounding areas remain large. We concluded that patches of vegetation after a forest fire are interesting to study, providing clues of why unburned patches occur despite the fire presence nearby, how they can be characterized spatially and how the vegetation composition responds to such nearby fire. The Recommendations for Resource Managers: Forest fires affect the forests, and have an effect on the population living within the forest and close to it. A forest fire commonly leaves behind a large number of unburnt vegetation patches. In this study we call them holes. These holes have been under severe heat and smoke pressure, but have similar tree species and forest structure as the original forest. They serve as the starting point to regenerate the forest. The primary implications for resource management are as follows: A better understanding of where they are, and how they are composed may help to understand the behavior of a fire. Their characterization may help to better understand the relation between vegetation as a fuel for forest fire. Their biodiversity will improve the fire spread modeling of burns that are carried out for management of a forest stand