Maxima of log-correlated fields: some recent developments*

We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.

Damage Detection and Localization under Variable Environmental Conditions Using Compressed and Reconstructed Bayesian Virtual Sensor Data

Structural health monitoring (SHM) with a dense sensor network and repeated vibration measurements produces lots of data that have to be stored. If the sensor network is redundant, data compression is possible by storing the signals of selected Bayesian virtual sensors only, from which the omitted signals can be reconstructed with higher accuracy than the actual measurement. The selection of the virtual sensors for storage is done individually for each measurement based on the reconstruction accuracy. Data compression and reconstruction for SHM is the main novelty of this paper. The stored and reconstructed signals are used for damage detection and localization in the time domain using spatial or spatiotemporal correlation. Whitening transformation is applied to the training data to take the environmental or operational influences into account. The first principal component of the residuals is used to localize damage and also to design the extreme value statistics control chart for damage detection. The proposed method was studied with a numerical model of a frame structure with a dense accelerometer or strain sensor network. Only five acceleration or three strain signals out of the total 59 signals were stored. The stored and reconstructed data outperformed the raw measurement data in damage detection and localization.

On the locations of maxima and minima in a sequence of exchangeable random variables

We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.

Evaluation of Fatigue Characteristics of Aluminum Alloys and Mechanical Components Using Extreme Value Statistics and C-specimens

In this study, the fatigue characteristics of aluminum alloys and mechanical components were investigated. To evaluate the effect of forging, fatigue specimens with the same chemical compositions were prepared from billets and forged mechanical components. To evaluate the cleanliness of the aluminum alloys, the cross-sectional area of specimens was observed, and the maximum inclusion sizes were obtained using extreme value statistics. Rotary bending fatigue tests were performed, and the fracture surfaces of the specimens were analyzed. The results show that the forging process not only elevated the fatigue strength but also reduced the scatter of the fatigue life of aluminum alloys. The fatigue characteristics of C-specimens were obtained to develop finite-element method (FEM) models. With the intrinsic fatigue properties and strain–life approach, the FEM analysis results agreed well with the test results.

The Use of Extreme Value Statistics to Characterize Blood Glucose Curves and Patient Level Risk Assessment of Patients With Type I Diabetes

Objective: Characterizing blood glucose curves and providing precise patient level risk assessment of hyperglycemia using extreme value statistics and comparing these assessments with traditional indicators of glycemic variability which are not designed to specifically capture the risk of hyperglycemia. Research Design and Methods: One year return level (blood glucose level exceeded exactly once every year on average) and probability of exceeding and expected time spent above certain thresholds (600 and 400 mg/dL) per year were calculated. As a comparison, traditional metrics for glycemic variability were determined too. The effect of body mass index on extremes was also investigated using non-stationary models. Metrics were calculated on a dataset containing 170.8 patient-years of measurements of 226 patients. Results: Nine high-risk patients were identified with the novel metrics: their estimated time spent above 600 mg/dL per year were above 2 hours. These patients were at moderate risk according to the traditional metrics. Higher body mass index was associated with more extreme glucose levels. Conclusions: Through these estimates it is possible to assess each patient’s individual clinical risk of hyperglycemia even beyond the observed blood glucose levels and detection limits. Additionally, it allows the assessment of the impact of clinical characteristics and treatments on blood glucose control in a novel, mathematically well-founded and potentially clinically more useful way than the already existing indicators.

Human mortality at extreme age

We use a combination of extreme value statistics, survival analysis and computer-intensive methods to analyse the mortality of Italian and French semi-supercentenarians. After accounting for the effects of the sampling frame, extreme-value modelling leads to the conclusion that constant force of mortality beyond 108 years describes the data well and there is no evidence of differences between countries and cohorts. These findings are consistent with use of a Gompertz model and with previous analysis of the International Database on Longevity and suggest that any physical upper bound for the human lifespan is so large that it is unlikely to be approached. Power calculations make it implausible that there is an upper bound below 130 years. There is no evidence of differences in survival between women and men after age 108 in the Italian data and the International Database on Longevity, but survival is lower for men in the French data.