Georg Lindgren

Test-Retest Variability in Glaucomatous Visual Fields

We measured test-retest variations in computerized visual fields from glaucomatous eyes. Fifty-one patients were tested four times within a four-week period; the severity of disease varied from incipient to advanced. We determined the dependence of threshold variability on defect depth and test point location. In areas of the visual field initially found to have moderate loss of sensitivity, variation in follow-up measurements ranged from normal sensitivity to absolute defect, with little dependence on distance from fixation. Conversely, large changes were considerably more unusual in locations initially showing normal or near-normal sensitivities, and variability was lowest in the most central portion of the field. Our findings suggest that differentiation between true progression and random variation will be facilitated if these factors are taken into account, as well as if comparisons are based on more than two tests. The complex nature of interest variation in glaucoma makes it natural to approach this problem with the help of computer-assisted analyses.

Point processes of exits by bivariate Gaussian processes and extremal theory for the chi^2-process and its concomitants

Let [zeta](t), [eta](t) be continuously differentiable Gaussian processes with mean zero, unit variance, and common covariance function r(t), and such that [zeta](t) and [eta](t) are independent for all t, and consider the movements of a particle with time-varying coordinates ([zeta](t), [eta](t)). The time and location of the exists of the particle across a circle with radius u defines a point process in R3 with its points located on the cylinder {(t, u cos [theta], u sin [theta]); t >= 0, 0 0 as t --> [infinity], the time and space-normalized point process of exits converges in distribution to a Poisson process on the unit cylinder. As a consequence one obtains the asymptotic distribution of the maximum of a [chi]2-process, [chi]2(t) = [zeta]2(t) + [eta]2(t), P{sup0 e-[tau] if T(-r(0)/2[pi])1/2u - exp(-u2/2) --> [tau] as T, u --> [infinity]. Furthermore, it is shown that the points in R3 generated by the local [epsilon]-maxima of [chi]2(t) converges to a Poisson process in R3 with intensity measure (in cylindrical polar coordinates) (2[pi]r2)-1 dt d[theta] dr. As a consequence one obtains the asymptotic extremal distribution for any function g([zeta](t), [eta](t)) which is almost quadratic in the sense that has a limit g*([theta]) as r --> [infinity]. Then P{sup0 exp(-([tau]/2[pi]) [integral operator] [theta] = 02[pi] e-g*([theta]) d[theta]) if T(-r(0)/2[pi])1/2u exp(-u2/2) --> [tau] as T, u --> [infinity].

Use and Structure of Slepian Model Processes for Prediction and Detection in Crossing and Extreme Value Theory

A Slepian model is a random function representation of the conditional behaviour of a Gaussian process after events defined by its level or curve crossings. It contains one regression term with random (non-Gaussian) parameters, describing initial values of derivatives etc. at the crossing, and one (Gaussian) residual process. Its explicit structure makes it well suited for probabilistic manipulations, finite approximations, and asymptotic expansions.

First Order Stochastic Lagrange Model for Asymmetric Ocean Waves

The Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century, is well understood, and there exist both exact theory and efficient numerical algorithms for calculation of the statistical distribution of wave characteristics. It is well suited for moderate seastates and deep water conditions. One drawback, however, is its lack of realism under extreme or shallow water conditions, in particular, its symmetry. It produces waves, which are stochastically symmetric, both in the vertical and in the horizontal direction. From that point of view, the Lagrangian wave model, which describes the horizontal and vertical movements of individual water particles, is more realistic. Its stochastic properties are much less known and have not been studied until quite recently. This paper presents a version of the first order stochastic Lagrange model that is able to generate irregular waves with both crest-trough and front-back asymmetries.

Basic Properties of Extremes and Level Crossings

We turn our attention now to continuous parameter stationary processes. We shall be especially concerned with stationary normal processes in this and most of the subsequent chapters but begin with a discussion of some basic properties which are relevant, whether or not the process is normal, and which will be useful in the discussion of extremal behaviour in later chapters.

Extremes of Continuous Parameter Stationary Processes

Our primary task in this chapter will be to discuss continuous parameter analogues of the sequence results of Chapter 3, and, in particular, to obtain a corresponding version of the Extremal Types Theorem which applies in the continuous parameter case. This will be taken up in the first section, using a continuous parameter analogue of the dependence restriction D(un). Limits for probabilities P{M(T) ≤ uT} are then considered for arbitrary families of constants {uT}, leading, in particular, to a determination of domains of attraction.

Maxima of Stationary Sequences

In this chapter, we extend the classical extreme value theory of Chapter 1 to apply to a wide class of dependent (stationary) sequences. The stationary sequences involved will be those exhibiting a dependence structure which is not “too strong”. Specifically, a distributional type of mixing condition— weaker than the usual forms of dependence restriction such as strong mixing—will be used as a basic assumption in the development of the theory.

Sample Path Properties at Upcrossings

Our main concern in the previous chapter has been the numbers and locations of upcrossings of high levels, and the relations between the upcrossings of several adjacent levels.For instance, we know from Theorem 9.3.2 and relation (9.2.3) that for a standard normal process each upcrossing of the high level u = uτ; with a probability p = τ*/τ is accompanied by an upcrossing also of the level n n

The Engineering Mathematics Study Programme in Lund: Background and Implementation

u_{tau * } = u - frac{{log p}} {u},