Robert J. Adler

The geometry of random fields

Preface to the Classics edition Preface Corrections and comments 1. Random fields and excursion sets 2. Homogeneous fields and their spectra 3. Sample function regularity 4. Geometry and excursion characteristics 5. Some expectations 6. Local maxima and high-level excursions 7. Some non-Gaussian fields 8. Sample function erraticism and Hausdorff dimension Appendix. The Markov property for Gaussian fields References Author index Subject index.

A practical guide to heavy tails: statistical techniques and applications

Part 1 Applications: heavy tailed probability distributions in the World Wide Web, M.E. Crovella et al self-similarity and heavy tails - structural modelling of network traffic, W. Willinger et al heavy tails in high-frequency financial data, U.A. Muller et al stable paretian modelling in finance - some empirical and theoretical aspects, S. Mittnik et al risk management and quantile estimation, F. Bassi et al. Part 2 Time series: analyzing stable time series, R.J. Adler et al inference for linear processes with stable noise, m. Calder, R.A. Davis on estimating the intensity of long-range dependence in finite and infinite variance time series, M.S. Taqqu, V. Teverovsky why non-linearities can ruin the heavy tailed modellers day, S.I. Resnick periodogram estimates from heavy-tailed data, T. Mikosch Bayesian inference for time series with infinite variance stable innovations, N. Ravishanker, Z. Qiou. Part 3 Heavy tail estimation: hill, bootstrap and jackknife estimators for heavy tails, O.V. Pictet et al characteristic function based estimation of stable distribution parameters, S.M. Kogan. D.B. Williams. Part 4 Regression: bootstrapping signs and permutations for regression with heavy tailed errors - a robust resampling, R. LePage et al linear regression with stable disturbances, J.H. McCulloch. Part 5 Signal processing: deviation from normality in statistical signal processing - parameter estimation with alpha-stable distributions, P. Tsakalides, C.L. Nikias statistical modelling and receiver design for multi-user communication networks, G.A. Tsihrintzis. Part 6 Model structures: subexponential distributions, C.M. Goldie, C. Kluppelberg structure of stationary stable processes, J. Rosinski tail behaviour of some shot noise processes, G. Samorodnitsky. Part 7 Numerical procedures: numerical approximation of the symmetric stable distribution and density, J.H. McCulloch table of the maximally-skewed stable distributions, J.H. McCulloch, D.B. Panton multivariate stable distributions - approximation, estimation, simulation and identification, J.P. Nolan univariate stable distributions -parametrizations and software, J.P. Nolan.

A NON-GAUSSIAN MODEL FOR RANDOM SURFACES

The central concern of this paper is to develop for rough (two-dimensional, metallic) surfaces a model other than the Gaussian one usually used. An analysis, via the notion of ‘upcrossing characteristics’, of some new data on abraded stainless steel, as well as a new look at some old data, indicates the need for such a model. The model adopted is of a form that gives X2-type marginal height distributions for the surface. After the new model has been introduced and motivated, its properties are investigated in some detail. In particular, the properties of the surface and its profiles at local maxima are studied by examining, for example, the height distribution and the surface curvature at such points. Phenomena are observed that are notably, qualitatively, different to what happens in the Gaussian model. Although the model introduced here is motivated by problems in the study of metallic surfaces, we believe it to be useful in other areas. Consequently, those sections of the paper that investigate the properties of the model are written so as to be independent of the original motivation. The paper also reintroduces, in an applied setting, the idea of examining surfaces via their upcrossing characteristics.

Stochastic modelling in physical oceanography

This volumes shows how modelling of oceanographic systems with probabilistic behaviour can lead to to an understanding of the way in which water interacts with the physical world.

Persistent Homology for Random Fields and Complexes

We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point cloud data, the algebraic structure of simplicial complexes determined by random vertices and, in most detail, the algebraic topology of the excursion sets of random elds.

Representations, decompositions and sample function continuity of random fields with independent increments

Standard fare in the study of representations and decompositions of processes with independent increments is pursued in the somewhat more complex setting of vector-valued random fields having independent increments over disjoint sets. Such processes are first constructed as almost surely uniformly convergent sums of Poisson type summands, that immediately yield information on sample function properties of versions. The constructions employed, which include a generalized version of the Ferguson-Klass construction with uniform convergence, are new even in the simpler setting of processes in one-dimensional time. Following these constructions, or representations, an analogue of the Levy-Ito decomposition for Levy processes is developed, which then enables a number of simple sample function properties of these processes to be read off from the Levy measure in their characteristic functionals. The paper concludes with a study of general centred additive random fields and an appendix incorporating a brief survey of the theory of centred sums of independent random variables.

MULTIPLE TESTING OF LOCAL MAXIMA FOR DETECTION OF PEAKS IN 1D

A topological multiple testing scheme for one-dimensional domains is proposed where, rather than testing every spatial or temporal location for the presence of a signal, tests are performed only at the local maxima of the smoothed observed sequence. Assuming unimodal true peaks with finite support and Gaussian stationary ergodic noise, it is shown that the algorithm with Bonferroni or Benjamini-Hochberg correction provides asymptotic strong control of the family wise error rate and false discovery rate, and is power consistent, as the search space and the signal strength get large, where the search space may grow exponentially faster than the signal strength. Simulations show that error levels are maintained for nonasymptotic conditions, and that power is maximized when the smoothing kernel is close in shape and bandwidth to the signal peaks, akin to the matched filter theorem in signal processing. The methods are illustrated in an analysis of electrical recordings of neuronal cell activity.

Extrema and level crossings of χ2 processes

Abstract : This document studies the sample path behavior of sq X processes in the neighbourhood of their level crossings and extreme via the development of Slepian model processes. The results, aside from being of particular interest in the study of sq X processes, have a general interest insofar as they indicate which properties of Gaussian processes (which has been heavily researched in this regard) are mirrored or lost when the assumption of normality is not made. Particular emphasis is placed on the behavior of sq X processes at both high and low levels, these being of considerable practical importance. Also extended are previous results on the asymptotic Poisson form of the point process of high maxima to include also low minima (which are in a different domain of attraction) thus closing a gap in the theory of 59 sq X processes. Keywords: Poisson limit; Stochastic processes.

High level excursion set geometry for non-Gaussian infinitely divisible random fields

NSA grant MSPF-05G-049 ARO grant W911NF-07-1-0078 US-Israel Binational Science Foundation, grant 2004064

Topological complexity of smooth random functions

1 Introduction.- 2 Gaussian Processes.- 3 Some Geometry and Some Topology.- 4 The Gaussian Kinematic Formula.- 5 On Applications: Topological Inference.- 6 Algebraic Topology of Excursion Sets: A New Challenge