The Clifford-Fourier Transform {\mathcal {F}_o} and Monogenic Extensions

In the last decade several versions of the Fourier transform have been formulated in the framework of Clifford algebra. We present a (Clifford-Fourier) transform, constructed using the geometric properties of Clifford algebra. We show the corresponding results of operational calculus, and a connection between the Fourier transform and this new transform. We obtain a technique to construct monogenic extensions of a certain type of continuous functions, and versions of the Paley-Wiener theorems are formulated.     

On the simplified pair-copula construction — Simply useful or too simplistic?

Due to their high flexibility, yet simple structure, pair-copula constructions (PCCs) are becoming increasingly popular for constructing continuous multivariate distributions. However, inference requires the simplifying assumption that all the pair-copulae depend on the conditioning variables merely through the two conditional distribution functions that constitute their arguments, and not directly. In terms of standard measures of dependence, we express conditions under which a specific pair-copula decomposition of a multivariate distribution is of this simplified form. Moreover, we show that the simplified PCC in fact is a rather good approximation, even when the simplifying assumption is far from being fulfilled by the actual model.     

Smoothness in Bayesian Non-parametric Regression with Wavelets

Consider the classical nonparametric regression problem y_i = f(t_i) + _ii = 1,...,n where t_i = i/n, and _i are i.i.d. zero mean normal with variance ². The aim is to estimate the true function f which is assumed to belong to the smoothness class described by the Besov space B _pq ^q . These are functions belonging to L_p with derivatives up to order s, in L_p sense. The parameter q controls a further finer degree of smoothness. In a Bayesian setting, a prior on B _pq ^q is chosen following Abramovich, Sapatinas and Silverman (1998). We show that the optimal Bayesian estimator of f is then also a.s. in B _pq ^q if the loss function is chosen to be the Besov norm of B _pq ^q . Because it is impossible to compute this optimal Bayesian estimator analytically, we propose a stochastic algorithm based on an approximation of the Bayesian risk and simulated annealing. Some simulations are presented to show that the algorithm performs well and that the new estimator is competitive when compared to the more standard posterior mean.     

Covariate Selection in High-Dimensional Generalized Linear Models With Measurement Error

In many problems involving generalized linear models, the covariates are subject to measurement error. When the number of covariates p exceeds the sample size n, regularized methods like the lasso or Dantzig selector are required. Several recent papers have studied methods which correct for measurement error in the lasso or Dantzig selector for linear models in the p > n setting. We study a correction for generalized linear models, based on Rosenbaum and Tsybakov’s matrix uncertainty selector. By not requiring an estimate of the measurement error covariance matrix, this generalized matrix uncertainty selector has a great practical advantage in problems involving high-dimensional data. We further derive an alternative method based on the lasso, and develop efficient algorithms for both methods. In our simulation studies of logistic and Poisson regression with measurement error, the proposed methods outperform the standard lasso and Dantzig selector with respect to covariate selection, by reducing the number of false positives considerably. We also consider classification of patients on the basis of gene expression data with noisy measurements. Supplementary materials for this article are available online.     

A Dynamic Mixture Model for Unsupervised Tail Estimation without Threshold Selection

Exceedances over high thresholds are often modeled by fitting a generalized Pareto distribution (GPD) on R⁺. It is difficult to select the threshold, above which the GPD assumption is enough solid and enough data is available for inference. We suggest a new dynamically weighted mixture model, where one term of the mixture is the GPD, and the other is a light-tailed density distribution. The weight function varies on R⁺ in such a way that for large values the GPD component is predominant and thus takes the role of threshold selection. The full data set is used for inference on the parameters present in the two component distributions and in the weight function. Maximum likelihood provides estimates with approximate standard deviations. Our approach has been successfully applied to simulated data and to the (previously studied) Danish fire loss data set. We compare the new dynamic mixture method to Dupuis' robust thresholding approach in peaks-over-threshold inference. We discuss robustness with respect to the choice of the light-tailed component and the form of the weight function. We present encouraging simulation results that indicate that the new approach can be useful in unsupervised tail estimation, especially in heavy tailed situations and for small percentiles.     

Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study

Estimation of flood and drought frequencies is important for reservoir design and management, river pollution, ecology and drinking water supply. Through an example based on daily streamflow observations, we introduce a stepwise procedure for estimating quantiles of the hydrological extremes floods and droughts. We fit the generalised extreme value (GEV) distribution by the method of block maxima and the generalised Pareto (GP) distribution by applying the peak over threshold method. Maximum likelihood, penalized maximum likelihood and probability weighted moments are used for parameter estimation. We incorporate trends and seasonal variation in the models instead of splitting the data, and investigate how the observed number of extreme events, the chosen statistical model, and the parameter estimation method effect parameter estimates and quantiles. We find that a seasonal variation should be included in the GEV distribution fitting for floods using block sizes less than one year. When modelling droughts, block sizes of one year or less are not recommended as significant model bias becomes visible. We conclude that the different characteristics of floods and droughts influence the choices made in the extreme value modelling within a common inferential strategy.This revised version was published online in March 2005 with corrections to the cover date.     

Quantitative model calculation of the time-dependent protoporphyrin IX concentration in normal human epidermis after delivery of ALA by passive topical application or lontophoresis

We present a mathematical layer model to quantitatively calculate the diffusion of 5-aminolevulinic acid (ALA) in the skin in vivo, its uptake into the cells and its conversion to protoporphyrin IX (PpIX) and subsequently to heme. The model is a modification and extension of a recently presented three-compartment model. The diffusion of ALA in the skin (epidermis, dermis) is described by the time-dependent diffusion equation, and the sink in this equation accounts for ALA uptake in the cells. As boundary conditions, we use the ALA flux across the human stratum corneum (SC) in vitro during passive or iontophoretic ALA delivery as measured in vitro. Besides the diffusion equation, the model includes three additional equations, similar in form to those of the three-compartment model but with a different interpretation. Our additional equations are supposed to describe, respectively, the conversion of ALA in the cytoplasm to some intermediate compound in the mitochondria and the conversion of the latter to PpIX and of PpIX to heme. The first conversion is a process of the Michaelis-Menten type, the other two are first-order rate processes. When fitted to the published data of PpIX fluorescence from normal human skin following iontophoresis of ALA, the model yields the tissue concentration of PpIX as a function of time after ALA application. The computed concentrations are in good agreement with the published phototoxic concentrations of PpIX in the tissues obtained from extraction. The model parameters obtained from the fit are subsequently used to compute the PpIX concentration in normal human skin after 4 h topical application of 10, 20 and 40% ALA. This again yields the PpIX concentrations in tissue, in good agreement with the published values. The saturation of the PpIX concentration as a function of applied ALA concentration is calculated and agrees with clinical observations on the effectiveness of photodynamic therapy. Photobleaching is simulated, with subsequent resynthesis of PpIX in qualitative agreement with experiment. Finally, the model predicts that only 2.5-3.5% of the ALA entering the skin after passing the SC is converted to PpIX. The layered model is a considerable simplification of real skin, but its successful qualitative and quantitative reproduction of experimental data may encourage further studies to test and refine the model to improve our understanding of the kinetics of ALA and the synthesis of PpIX in the skin.     

Eine Methode zur Analyse des K?ses

Ohne Zusammenfassung     

Alkaloids from Eucharis amazonica (Amaryllidaceae)

Thirteen alkaloids have been isolated from dried bulbs and leaves of flowering Eucharis amazonica (Amaryllidaceae). The alkaloids, 7-methoxyoxoassoanine, 6-O-methylpretazettine and apohaemanthamine, are reported for the first time from a natural source.     

A dynamical phase transition in a caricature of a spin glass

This paper studies the rate of convergence to equilibrium of Glauber dynamics (Gibbs Sampler) for a system ofN Ising spins with random energy (at inverse temperature >0). For each of the 2 ^N spin configurations the energy is drawn independently from the values 0 and-logN with probabilities 1-N ^–7, resp.N ^– (>0), and is kept fixed during the evolution. The main result is an estimate of the coupling time of two Glauber dynamics starting from different configurations and coupled via the same updating noise. AsN the system exhibits two dynamical phase transitions: (1) at =1 the coupling time changes from polynomial (>1) to stretched exponential (<1) inN; (2) if <1, then at = the almost coupling time [i.e., the first time that the two dynamics are within distanceo(N)] changes from polynomial (<) to stretched exponential (>) inN. The techniques used to control the randomness in the coupling are static and dynamic large-deviation estimates and stochastic domination arguments.