Currently popular techniques such as experimental spectroscopy and computer-aided molecular modelling lead to data having very many variables observed on each of relatively few individuals. A common objective is discrimination between two or more groups, but the direct application of standard discriminant methodology fails because of singularity of covariance matrices. The problem has been circumvented in the past by prior selection of a few transformed variables, using either principal component analysis or partial least squares. Although such selection ensures nonsingularity of matrices, the decision process is arbitrary and valuable information on group structure may be lost. We therefore consider some ways of estimating linear discriminant functions without such prior selection. Several spectroscopic data sets are analysed with each method, and questions of bias of assessment procedures are investigated. All proposed methods seem worthy of consideration in practice.
Abstract:Single-pixel detectors can be used as imaging devices by making use of structured illumination. These systems work by correlating a changing incident light field with signals measured on a photodiode to derive an image of an object. In this work we demonstrate a system that utilizes a digital light projector to illuminate a scene with approximately 1300 different light patterns every second and correlate these with the back scattered light measured by three spectrally-filtered single-pixel photodetectors to produce a full-color high-quality image in a few seconds of data acquisition. We utilize a differential light projection method to self normalize the measured signals, improving the reconstruction quality whilst making the system robust to external sources of noise. This technique can readily be extended for imaging applications at non-visible wavebands.
In environmental applications it is common for the extremes of a variable to be non-stationary, varying systematically in space, time or with the values of covariates. Multi-site datasets are common, and in such cases there is likely to be non-negligible inter-site dependence. We consider applications in which multi-site data are used to infer the marginal behaviour of the extremes at individual sites, while adjusting for inter-site dependence. For reasons of statistical efficiency, it is standard to model exceedances of a high threshold. Choosing an appropriate threshold can be problematic, particularly if the extremes are non-stationary. We propose a method for setting a covariate-dependent threshold using quantile regression. We consider how the quantile regression model and extreme value models fitted to threshold exceedances should be parameterized, in order that they are compatible. We adjust estimates of uncertainty for spatial dependence using methodology proposed recently. These methods are illustrated using time series of storm peak significant wave heights from 72 sites in the Gulf of Mexico. A simulation study illustrates the applicability of the proposed methodology more generally.
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