http://library.um.ac.id/free-contents/downloadpdf.php/buku/laws-of-small-numbers-extremes-and-rare-events-michael-falk-jurg-husler-rolf-dieter-reiss-42718.pdf

Laws of small numbers: extremes and rare events / Michael Falk, Jurg Husler, Rolf-Dieter Reiss

PURPOSE. To evaluate a semiautomated image analysis software package, Retinal Image multiScale Analysis (RISA), for the diagnosis of plus disease in preterm infants with retinopathy of prematurity (ROP). METHODS. Digital images of the posterior pole showing both disc and macula in preterm infants with ROP were analyzed with an enhanced version of RISA. Venules (N 106) and arterioles (N 44) were identified, and integrated curvature, diameter, and tortuosity of the vessels were calculated. After the RISA calculations were completed, the origins of the vessels were determined to be 32 eyes in 16 infants (12 eyes with plus disease, 20 with no plus disease, as diagnosed by ophthalmic examination). Vessels were sorted into two groups—plus disease and no plus disease—and each RISA parameter was compared using the Mann-Whitney test. For each parameter, sensitivity and specificity were plotted as a function of cutoff criterion, receiver operating characteristic (ROC) curves were constructed, and the areas under the curve (AUC) were calculated. RESULTS. For both arterioles and venules, each of the three parameters was significantly larger for the plus disease group. For instance, the median estimated arteriolar and venular diameters were approximately 12 m greater in plus disease. Sensitivity and specificity plots indicated good accuracy of each parameter for the diagnosis of plus disease. The AUC showed that curvature had the highest diagnostic accuracy (0.911 for arterioles, 0.824 for venules). CONCLUSIONS. The strong performance of RISA parameters in this sample suggests that RISA may be useful for diagnosing plus disease in preterm infants with ROP. (Invest Ophthalmol Vis Sci. 2005;46:4734‐4738) DOI:10.1167/iovs.05-0646

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Diagnosis of plus disease in retinopathy of prematurity using Retinal Image multiScale Analysis.

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area of extreme events, underlining its relevance for mathematics, natural sciences, engineering, and social sciences. After exploring the basics of the classical theory of extreme events, the book presents a careful examination of how a dynamical system can serve as a generator of stochastic processes, and explores in detail the relationship between the hitting and return time statistics of a dynamical system and the possibility of constructing extreme value laws for given observables. Explicit derivation of extreme value laws are then provided for selected dynamical systems. The book then discusses how extreme events can be used as probes for inferring fundamental dynamical and geometrical properties of a dynamical system and for providing a novel point of view in problems of physical and geophysical relevance. A final summary of the main results is then presented along with a discussion of open research questions. Finally, an appendix with software in Matlab programming language allows the readers to develop further understanding of the presented concepts.

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Extremes and Recurrence in Dynamical Systems

https://www.reading.ac.uk/web/FILES/maths/AnaFreitascom_Reading_2013_6.pdf

The extremal index, hitting time statistics and periodicity

The article develops the approach of Ferro and Segers (J.R. Stat. Soc., Ser. B 65:545, 2003) to the estimation of the extremal index, and proposes the use of a new variable decreasing the bias of the likelihood based on the point process character of the exceedances. Two estimators are discussed: a maximum likelihood estimator and an iterative least squares estimator based on the normalized gaps between clusters. The first provides a flexible tool for use with smoothing methods. A diagnostic is given for condition \(D^{(2)}(u_n)\), under which maximum likelihood is valid. The performance of the new estimators were tested by extensive simulations. An application to the Central England temperature series demonstrates the use of the maximum likelihood estimator together with smoothing methods.

Likelihood estimation of the extremal index

A local mixing condition D(k) is introduced for stationary sequences satisfying Leadbetter's condition D. Under the local mixing condition, the asymptotic distribution of the sample maximum can be calculated with the knowledge of the joint distribution of k consecutive terms. Some examples are given to illustrate the notion.

Calculating the extremal index for a class of stationary sequences

Consider the shot noise process X(t):= .ih(t - rj), t---0, where h is a bounded positive non-increasing function supported on a finite interval, and the r;'s are the points of a renewal process tj on [0, oo). In this paper, the extremal properties of (X(t)} are studied. It is shown that these properties can be investigated in a natural way through a discrete-time process which records the states of (X(t)} at the points of ti. The important special case where 7t is Poisson is treated in detail, and a domain-of-attraction result for the compound Poisson distribution is obtained as a by-product. EXTREME VALUE; POINT PROCESS; RENEWAL PROCESS

Extremal properties of shot noise processes

The crossing intensity of a level by a shot noise process with a monotone response is studied, and it is shown that the intensity can be naturally expressed in terms of a marginal probability.

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Tailen Hsing

Papers

Calculating the extremal index for a class of stationary sequences

Extremal properties of shot noise processes

On the intensity of crossings by a shot noise process