There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Statistics for Spatial Data.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

An overview of model-based geostatistics.- Gaussian models for geostatistical data.- Generalized linear models for geostatistical data.- Classical parameter estimation.- Spatial prediction.- Bayesian inference.- Geostatistical design.

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Model-based Geostatistics

Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

https://rss.onlinelibrary.wiley.com/doi/epdf/10.1111/j.1467-9868.2011.00777.x

An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach

A new definition of the rainflow cycle counting method

iii Foreword This is a tutorial for how to use the MATLAB toolbox WAFO for analysis and simulation of random waves and random fatigue. The toolbox consists of a number of MATLAB m-files together with executable routines from FORTRAN or C++ source, and it requires only a standard MATLAB setup, with no additional toolboxes. A main and unique feature of WAFO is the module of routines for computation of the exact statistical distributions of wave and cycle characteristics in a Gaussian wave or load process. The routines are described in a series of examples on wave data from sea surface measurements and other load sequences. There are also sections for fatigue analysis and for general extreme value analysis. Although the main applications at hand are from marine and reliability engineering, the routines are useful for many other applications of Gaussian and related stochastic processes. The routines are based on algorithms for extreme value and crossing analysis, developed over many years by the authors as well as many results available in the literature. References are given to the source of the algorithms whenever it is possible. These references are given in the MATLAB-code for all the routines and they are also listed in the last section of this tutorial. If the references are not used explicitly in the tutorial; it means that it is referred to in one of the MATLAB m-files. Besides the dedicated wave and fatigue analysis routines the toolbox contains many statistical simulation and estimation routines for general use, and it can therefore be used as a toolbox for statistical work. These routines are listed, but not explicitly explained in this tutorial. The present toolbox represents a considerable development of two earlier toolboxes, the FAT and WAT toolboxes, for fatigue and wave analysis, respectively. These toolboxes were both Version 1; therefore WAFO has been named Version 2. The routines in the tutorial are tested on WAFO-version 2.5, which was made available in beta-version in January 2009 and in a stable version in February 2011. The persons that take actively part in creating this tutorial are (in alphabetical order): Per v vi Many other people have contributed to our understanding of the problems dealt with in this text, first of all Professor Ross Leadbetter at the University of North Carolina at Chapel Hill and Professor Krzysztof Podgórski, Mathematical Statistics, Lund University. We would also like to particularly thank Michel …

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WAFO - A Matlab Toolbox For Analysis of Random Waves And Loads

In this paper we discuss cycle counting methods, such as rainflow-, crest-to-trough-, positive peak-count and different damage accumulation rules, for irregular random loads, which have an infinite number of local extremes in finite intervals, e.g. the fourth spectral moment is infinite. We present conservative bounds for the expected damage for such loads from the upcrossing intensity. These results are illustrated by examples of Gaussian, xz-, and Morison-loads. NOMENCLATURE E(X) = mathematical expectation of the random variable X E(X I Y = y) = conditional expectation of X given Y = y Var(X) = variance of the random variable X a E A = a is an element of the set A {f; . } = the set of points t fulfilling the condition . # { . } = the number of elements in the set { . } 1 {. ,(x) = the indicator function of the set { . } sup{ . } = the supremum (least upper bound) of elements in the set { . } inf{ } = the infimum (greatest lower bound) of elements in the set { . 1

Note on cycle counts in irregular loads

On the 'narrow-band' approximation for expected fatigue damage

This study focuses on the statistical description and analysis of road surface irregularities that are essential for heavy-vehicle fatigue assessment. Three new road profile models are proposed: a homogenous Laplace moving average process, a non-homogenous Laplace process and a hybrid model that combines Gaussian and Laplace modelling. These are compared with the classical homogenous Gaussian process as well as with the non-homogenous Gaussian model that represents the road surface as a homogenous Gaussian process with Motor Industry Research Association spectrum enhanced by randomly placed and shaped irregularities. The five models are fitted to eight measured road surfaces and their accuracy and efficiency are discussed.

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Igor Rychlik

Papers

A new definition of the rainflow cycle counting method

WAFO - A Matlab Toolbox For Analysis of Random Waves And Loads

Note on cycle counts in irregular loads

On the 'narrow-band' approximation for expected fatigue damage

Models for road surface roughness