Outlier robust corner-preserving methods for reconstructing noisy images

The ability to remove a large amount of noise and the ability to preserve most structure are desirable properties of an image smoother. Unfortunately, they usually seem to be at odds with each other; one can only improve one property at the cost of the other. By combining M-smoothing and least-squares-trimming, the TM-smoother is introduced as a means to unify corner-preserving properties and outlier robustness. To identify edge- and corner-preserving properties, a new theory based on differential geometry is developed. Further, robustness concepts are transferred to image processing. In two examples, the TM-smoother outperforms other corner-preserving smoothers. A software package containing both the TM- and the M-smoother can be downloaded from the Internet.Comment: Published at http://dx.doi.org/10.1214/009053606000001109 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Trimmed likelihood estimators for stochastic differential equations with an application to crack growth analysis from photos

We introduce trimmed likelihood estimators for processes given by a stochastic differential equation for which a transition density is known or can be approximated and present an algorithm to calculate them. To measure the fit of the observations to a given stochastic process, two performance measures based on the trimmed likelihood estimator are proposed. The approach is applied to crack growth data which are obtained from a series of photos by backtracking large cracks which were detected in the last photo. Such crack growth data are contaminated by several outliers caused by errors in the automatic image analysis. We show that trimming 20% of the data of a growth curve leads to good results when 100 obtained crack growth curves are fitted with the Ornstein-Uhlenbeck process and the Cox-Ingersoll-Ross processes while the fit of the Geometric Brownian Motion is significantly worse. The method is sensitive in the sense that crack curves obtained under different stress conditions provide significantly different parameter estimates

Optimal designs for inspection times of interval-censored data

We treat optimal equidistant and optimal non-equidistant inspection times for interval-censored data with exponential distribution.We provide in particular a recursive formula for calculating the optimal non-equidistant inspection times which is similar to a formula for optimal spacing of quantiles for asymptotically best linear estimates based on order statistics. This formula provides an upper bound for the standardized Fisher information which is reached for the optimal non-equidistant inspection times if the number of inspections is converging to infinity. The same upper bound is also shown for the optimal equidistant inspection times. Since optimal equidistant inspection times are easier to calculate and easier to handle in practice, we study the efficiency of optimal equidistant inspection times with respect to optimal nonequidistant inspection times. Moreover, since the optimal inspection times are only locally optimal, we provide also some results concerning maximin efficient designs

Relative importance of fertiliser addition to plants and exclusion of predators for aphid growth in the field

Herbivore dynamics and community structure are influenced both by plant quality and the actions of natural enemies. A factorial experiment manipulating both higher and lower trophic levels was designed to explore the determinants of colony growth of the aphid Aphis jacobaeae, a specialist herbivore on ragwort Senecio jacobaea. Potential plant quality was manipulated by regular addition of NPK-fertiliser and predator pressure was reduced by interception traps; the experiment was carried out at two sites. The size and persistence of aphid colonies were measured. Fertiliser addition affected plant growth in only one site, but never had a measurable effect on aphid colony growth. In both habitats the action of insect predators dominated, imposing strong and negative effects on aphid colony performance. Ants were left unmanipulated in both sites and their performance on the aphid colonies did not significantly differ between sites or between treatments. Our results suggest that, at least for aphid herbivores on S. jacobaea, the action of generalist insect predators appears to be the dominant factor affecting colony performance and can under certain conditions even improve plant productivit

An overnational cereal circuit for developing locally adapted organic seeds of wheat

On several locations in Germany and Switzerland, new and local varieties of winter wheat were compared to the variety "Bussard" in trial plots with 2-4 replications. Among other parameters, baking quality and gluten content were analised and discussed

Prediction of crack growth based on a hierarchical diffusion model

A general Bayesian approach for stochastic versions of deterministic growth models is presented to provide predictions for crack propagation in an early stage of the growth process. To improve the prediction, the information of other crack growth processes is used in a hierarchical (mixed-effects) model. Two stochastic versions of a deterministic growth model are considered. One is a nonlinear regression setup where the trajectory is assumed to be the solution of an ordinary differential equation with additive errors. The other is a diffusion model defined by a stochastic differential equation (SDE) where increments have additive errors. Six growth models in the two versions are compared with respect to their ability to predict the crack propagation in a large data example. Two of them are based on the classical Paris-Erdogan law for crack growth, and four are other widely used growth models. It turned out that the three-parameter Paris-Erdogan model and the Weibull model provide the best results followed by the logistic model. Suprisingly, the SDE approach has no advantage for the prediction compared with the nonlinear regression setup

Simplified simplicial depth for regression and autoregressive growth processes

We simplify simplicial depth for regression and autoregressive growth processes in two directions. At first we show that often simplicial depth reduces to counting the subsets with alternating signs of the residuals. The second simplification is given by not regarding all subsets of residuals. By consideration of only special subsets of residuals, the asymptotic distributions of the simplified simplicial depth notions are normal distributions so that tests and confidence intervals can be derived easily. We propose two simplifications for the general case and a third simplification for the special case where two parameters are unknown. Additionally, we derive conditions for the consistency of the tests. We show that the simplified depth notions can be used for polynomial regression, for several nonlinear regression models, and for several autoregressive growth processes. We compare the efficiency and robustness of the different simplified versions by a simulation study concerning the Michaelis-Menten model and a nonlinear autoregressive process of order one

Tests based on simplicial depth for AR (1) models with explosion

We propose an outlier robust and distributions-free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. Using this, it is shown that the asymptotic distribution of the test statistic is given by an integrated two-dimensional Gaussian process. Conditions for the consistency of the test are given and the power of the test at finite samples is compared with five alternative tests, using errors with normal distribution, contaminated normal distribution, and Frechet distribution in a simulation study. The comparisons show that the new test outperforms all other tests in the case of skewed errors and outliers. Although here we deal with the AR(1) model with intercept only, the asymptotic results hold for any simplicial depth which reduces to alternating signs of three residuals

Bayesian prediction for a jump diffusion process with application to crack growth in fatigue experiments

In many felds of technological developments, understanding and controlling material fatigue is an important point of interest. This article is concerned with statistical modeling of the damage process of prestressed concrete under low cyclic load. A crack width process is observed which exhibits jumps with increasing frequency. Firstly, these jumps are modeled using a Poisson process where two intensity functions are presented and compared. Secondly, based on the modeled jump process, a stochastic process for the crack width is considered through a stochastic differential equation (SDE). It turns out that this SDE has an explicit solution. For both modeling steps, a Bayesian estimation and prediction procedure is presented

Prediction intervals for the failure time of prestressed concrete beams

The aim is the prediction of the failure time of prestressed concrete beams under low cyclic load. Since the experiments last long for low load, accelerated failure tests with higher load are conducted. However, the accelerated tests are expensive so that only few tests are available. To obtain a more precise failure time prediction, the additional information of time points of breakage of tension wires is used. These breakage time points are modeled by a nonlinear birth process. This allows not only point prediction of a critical number of broken tension wires but also prediction intervals which express the uncertainty of the prediction