High-breakdown linear discriminant analysis

The classification rules of linear discriminant analysis are defined by the true mean vectors and the common covariance matrix of the populations from which the data come. Because these true parameters are generally unknown, they are commonly estimated by the sample mean vector and covariance matrix of the data in a training sample randomly drawn from each population. However, these sample statistics are notoriously susceptible to contamination by outliers, a problem compounded by the fact that the outliers may be invisible to conventional diagnostics. High-breakdown estimation is a procedure designed to remove this cause for concern by producing estimates that are immune to serious distortion by a minority of outliers, regardless of their severity. In this article we motivate and develop a high-breakdown criterion for linear discriminant analysis and give an algorithm for its implementation. The procedure is intended to supplement rather than replace the usual sample-moment methodology of discriminant analysis either by providing indications that the dataset is not seriously affected by outliers (supporting the usual analysis) or by identifying apparently aberrant points and giving resistant estimators that are not affected by them.

Breast cancer five-year survival, by New South Wales regions, 1980 to 1991

Breast cancer five-year relative survival was calculated for 16 urban and rural regions in New South Wales (NSW) for cases incident in 1980-1991. Survival analysis employed cancer registry data linked with the death register, and age- and period-matched regional mortality of NSW women, Proportional hazard regression analysis was used to compare excess mortality in breast cancer cases in each region. The effect of region was significant (P < 0.05) in the analysis, after age and the follow-up variable (and their intel action) were adjusted for, although no region was significantly different from the referent group (chosen because of average relative five-year survival). When degree of spread and its interactions were entered into che model, the effect of region became nonsignificant. A significant linear trend (P < 0.05) in the adjusted relative risk for excess mortality in breast cancer cases was noted when regions were divided into quartiles based on socioeconomic status, with higher relative risk in low-socioeconomic-status groups; this effect also disappeared with adjustment for degree of spread at diagnosis. There was no general effect of rurality versus capital city or other metropolitan centres. This study demonstrates a small effect of region of residence and implied socioeconomic status on breast cancer survival in NSW women, but this becomes nonsignificant when the data are adjusted for degree of spread at diagnosis, This suggests that earlier diagnosis would he of benefit in reducing minor inequalities in breast cancer survival in NSW women.

Applying linear time-varying constraints to econometric models: With an application to demand systems

When linear equality constraints are invariant through time they can be incorporated into estimation by restricted least squares. If, however, the constraints are time-varying, this standard methodology cannot be applied. In this paper we show how to incorporate linear time-varying constraints into the estimation of econometric models. The method involves the augmentation of the observation equation of a state-space model prior to estimation by the Kalman filter. Numerical optimisation routines are used for the estimation. A simple example drawn from demand analysis is used to illustrate the method and its application.

Conformational dynamics of thyroid hormones by variable temperature nuclear magnetic resonance: The role of side chain rotations and cisoid/transoid interconversions

H-1 NMR spectra of the thyroid hormone thyroxine recorded at low temperature and high field show splitting into two peaks of the resonance due to the H2,6 protons of the inner (tyrosyl) ring. A single resonance is observed in 600 MHz spectra at temperatures above 185 K. An analysis of the line shape as a function of temperature shows that the coalescence phenomenon is due to an exchange process with a barrier of 37 kJ mol(-1). This is identical to the barrier for coalescence of the H2',6' protons of the outer (phenolic) ring reported previously for the thyroid hormones and their analogues. It is proposed that the separate peaks at low temperature are due to resonances for H2,6 in cisoid and transoid conformers which are populated in approximately equal populations. These two peaks are averaged resonances for the individual H2 and H6 protons. Conversion of cisoid to transoid forms can occur via rotation of either the alanyl side chain or the outer ring, from one face of the inner ring to the other. It is proposed that the latter process is the one responsible for the observed coalescence phenomenon. The barrier to rotation of the alanyl side chain is greater than or equal to 37 kJ mol(-1), which is significantly larger than has previously been reported for Csp(2)-Csp(3) bonds in other Ph-CH2-X systems. The recent crystal structure of a hormone agonist bound to the ligand-binding domain of the rat thyroid hormone receptor (Wagner et al. Nature 1995, 378, 690-697) shows the transoid form to be the bound conformation. The significant energy barrier to cisoid/transoid interconversion determined in the current study combined with the tight fit of the hormone to its receptor suggests that interconversion between the forms cannot occur at the receptor site but that selection for the preferred bound form occurs from the 50% population of the transoid form in solution.

Anisotropy-based performance analysis of linear discrete time invariant control systems

The anisotropic norm of a linear discrete-time-invariant system measures system output sensitivity to stationary Gaussian input disturbances of bounded mean anisotropy. Mean anisotropy characterizes the degree of predictability (or colouredness) and spatial non-roundness of the noise. The anisotropic norm falls between the H-2 and H-infinity norms and accommodates their loss of performance when the probability structure of input disturbances is not exactly known. This paper develops a method for numerical computation of the anisotropic norm which involves linked Riccati and Lyapunov equations and an associated special type equation.

A flux ratio theorem for multicomponent linear diffusion equations

Ussing [1] considered the steady flux of a single chemical component diffusing through a membrane under the influence of chemical potentials and derived from his linear model, an expression for the ratio of this flux and that of the complementary experiment in which the boundary conditions were interchanged. Here, an extension of Ussing's flux ratio theorem is obtained for n chemically interacting components governed by a linear system of diffusion-migration equations that may also incorporate linear temporary trapping reactions. The determinants of the output flux matrices for complementary experiments are shown to satisfy an Ussing flux ratio formula for steady state conditions of the same form as for the well-known one-component case. (C) 2000 Elsevier Science Ltd. All rights reserved.

Stability of linear difference and differential inclusions

Any given n X n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N x N matrix B of spectral radius p(B) arbitrarily close to p(A). A difference inclusion x(k+1) is an element of Ax(k), where A is a compact set of matrices, is asymptotically stable if and only if A can be extended to a set B of nonnegative matrices B with \ \B \ \ (1) < 1 or \ \B \ \ (infinity) < 1. Similar results are derived for differential inclusions.

Estimates of the real structured radius of stability of linear dynamic systems

A question is examined as to estimates of the norms of perturbations of a linear stable dynamic system, under which the perturbed system remains stable in a situation R:here a perturbation has a fixed structure.

Fossil trees in ancient fluvial channel deposits: evidence of seasonal and longer-term climatic variability

It has been established that large numbers of certain trees can survive in the beds of rivers of northeastern Australia where a strongly seasonal distribution of precipitation causes extreme variations in flow on both a yearly and longer-term basis. In these rivers, minimal flow occurs throughout much of any year and for periods of up to several years, allowing the trees to become established and to adapt their form in order to facilitate their survival in environments that experience periodic inundation by fast-flowing, debris-laden water. Such trees (notably paperbark trees of the angiosperm genus Melaleuca) adopt a reclined to prostrate, downstream-trailing habit, have a multiple-stemmed form, modified crown with weeping foliage, development of thick, spongy bark, anchoring of roots into firm to lithified substrates beneath the channel floor, root regeneration, and develop in flow-parallel, linear groves. Individuals from within flow-parallel, linear groves are preserved in situ within the alluvial deposit of the river following burial and death. Four examples of in situ tree fossils within alluvial channel deposits in the Permian of eastern Australia demonstrate that specialised riverbed plant communities also existed at times in the geological past. These examples, from the Lower Permian Carmila Beds, Upper Permian Moranbah Coal Measures and Baralaba Coal Measures of central Queensland and the Upper Permian Newcastle Coal Measures of central New South Wales, show several of the characteristics of trees described from modern rivers in northeastern Australia, including preservation in closely-spaced groups. These properties, together with independent sedimentological evidence, suggest that the Permian trees were adapted to an environment affected by highly variable runoff, albeit in a more temperate climatic situation than the modem Australian examples. It is proposed that occurrences of fossil trees preserved in situ within alluvial channel deposits may be diagnostic of environments controlled by seasonal and longer-term variability in fluvial runoff, and hence may have value in interpreting aspects of palaeoclimate from ancient alluvial successions. (C) 2001 Elsevier Science B.V. All rights reserved.

Noise-free scattering of the quantized electromagnetic field from a dispersive linear dielectric

We study the scattering of the quantized electromagnetic field from a linear, dispersive dielectric using the scattering formalism for quantum fields. The medium is modeled as a collection of harmonic oscillators with a number of distinct resonance frequencies. This model corresponds to the Sellmeir expansion, which is widely used to describe experimental data for real dispersive media. The integral equation for the interpolating field in terms of the in field is solved and the solution used to find the out field. The relation between the ill and out creation and annihilation operators is found that allows one to calculate the S matrix for this system. In this model, we find that there are absorption bands, but the input-output relations are completely unitary. No additional quantum-noise terms are required.